ANALYTICALGEOMETRY

1. The branch of science which has obtained the name of analytical geometry owes its origin to Descartes. It consists in the application of algebra to geometry, not merely by using algebraic symbols as the representatives of magnitude, but by employing these in such a manner that the position of a point is indicated by referring it to fixed lines or planes, just as the position of a point on the earth's surface is defined by referring it to the equator and the first meridian. For example, if we suppose the positions of two lines at right angles to one another to be fixed and known, then the position of any point in the same plane is sufficiently determined by its distances from these two lines, just as the

position of a place on a given map is denoted by its latitude and longitude.

This science, like that of ordinary geometry, is divided into two parts, plane and solid, or analytical geometry of two and of three dimensions.

2. As every line or area is supposed to be situated in one plane, we may, whenever it is convenient to do so, imagine that plane to be the paper before us, which, for convenience of phraseology, we may further suppose to be standing

upright, so that any one line of print is horizontal, whilst the successive lines lie vertically under each other. The simplest way, under these circumstances, in which the position of a point can be indicated, will be by its distances from two lines at right angles to each other, the one horizontal, the other vertical. If we know the positions of these lines, then we know the position of every point in this plane whose distances from these two lines have been given us. But it is frequently necessary to adopt lines of reference inclined to one another. In this case, the position of a point is determined by its two distances from these lines respectively, each measured in a direction parallel to the other. The lines of reference are called the co-ordinate axes; their point of intersection is called the origin; the distances of any other point from the lines, measured in the way stated above are called the co-ordinates of that point; of these one is called the abscissa, the other the ordinate, the former being usually the designation of the horizontal distance.

In this figure, OX, OY, at right angles or not, are called the co-ordinate axes; O the origin; PM, PN (or its equal OM), drawn parallel to OY and OX, the co-ordinates of the point P; OM the abscissa, PM the ordinate. The line OM is usually written by the letter $x$, and MP by $y$. We may thus form some idea of the meaning of the symbols by considering the letter $x$ as the abbreviation for the general word longitude, and the letter $y$ as that for latitude. Thus when the latitude and longitude are both specified, the point referred to is determined. If, on the other hand, there is stated merely a relation between them, such, for example, as that the latitude is equal to the longitude, it is clear that no one point is determined thereby, since the same property is applicable to an innumerable series of points. A little consideration will show that some line, straight or crooked, will pass through all such points. The relation, therefore, is said to determine this line, and when expressed in algebraic symbols, is called the equation to the line. Thus, for example, the relation, "that the abscissa is equal to the ordinate," or the longitude equal to the latitude, understood as above, indicates a straight line; and the corresponding equation, $x=y$, is the equation to a straight line.

3. The requirements of analysis demand that attention should be paid to the sign as well as to the magnitude of the algebraic symbol which expresses an abscissa or an ordinate. We lay it down as a rule, deduced at once from the algebraic definitions of the symbols $+$ and $-$, that if one of them denotes a line measured in one direction, the other will denote a line measured in the opposite direction. Thus if $x\; =\; +a$ be an abscissa measured to the right of O, $x\; =\; -a$ will be an equal abscissa measured to the left; if $y\; =\; +b$ be an ordinate measured above O, $y\; =\; -b$ will be an equal ordinate measured below it; and these are the directions which we shall usually assign to the positive and negative abscissas and ordinates.

4. Besides the method of representation which we have described, there is another very unlike it in form, but equally important in the solution of mechanical and astronomical problems. In this system, the position of a point is determined by its distance from a given point, and the direction in which the line measuring that distance lies with respect to a given line. This system is called the system of polar co-ordinates. The given point to which all others are referred is called the pole, and the distance from that point the radius vector.

5. The subject before us naturally divides itself into

three distinct branches,—1°, the determination of the equations to curves from the knowledge of their properties; 2°, the determination of the forms and properties of curves from their equations; and, 3°, the deduction of one property of a curve from another, or from several others. The third branch being the combination and extension of the other two, we propose briefly to illustrate the first and second, and then to include under the third a discussion of some of the more important properties of the Conic Sections in the form in which the requirements of science demand their exhibition.

SECTION I.—THE DETERMINATION OF THE EQUATIONS TO CURVES FROM THE KNOWLEDGE OF THEIR PROPERTIES.

6. a. The Straight Line.—The property of the straight line which we shall employ is this—that if two series of parallel lines be drawn from any points in it so as to form with the given line a series of triangles, the sides of these triangles will be proportional.

Let PQR be the straight line; OX, OY the co-ordinate axes; PM parallel to OY, and QE parallel to OX: the triangles PQE, QRO are similar, $\backslash therefore\; PE\; :\; EQ\; ::\; QO\; :\; OR$.

Let OM be called $x$, or $OM\; =\; x$, $MP\; =\; y$, $OQ\; =\; b$; and let the ratio $QO\; :\; OR$ be $a\; :\; 1$.

Hence it is clear that any simple equation between $y$ and $x$ will represent a straight line. If the axes of co-ordinates are at right angles to each other—in which case they are called rectangular axes—$a$ is the trigonometrical tangent of the angle PRO, in which the given line cuts the axis of $x$.

7. The equation to a straight line may be written in a different form, which is frequently convenient, thus:

Let the perpendicular from O on the line PR be called $p$, the angle PRM $\backslash alpha$; then it is evident that

which is the equation to the line.

This equation is abbreviated by writing for it simply $\backslash alpha\; =\; 0$; or, when multiplied by a constant $a$, by $u\; =\; 0$; where $u$ is a function of $x$, $y$, $\backslash alpha$, $p$, and a constant which is arbitrary. We shall give an instance of the use of this form of the equation in the sequel.

If the axes of co-ordinates are not at right angles, the equation becomes slightly changed in form.

Let the line make the angles $\backslash theta$ and $\backslash phi$ with the axes of $x$ and $y$ respectively; then

consequently all lines which are parallel to the given line, or for which $\backslash theta$ and $\backslash phi$ are the same, have the relative magnitudes of the co-efficients of $x$ and $y$ the same, and vice versa.

8. COR. 1.—If the line pass through a given point of which the co-ordinates are $x\text{'}$, $y\text{'}$, we shall have $y\; =\; ax\; +\; b$ generally,

or $b$ is determined.

COR. 2.—If the line pass through a second given point whose co-ordinates are $x\text{'}\text{'}$, $y\text{'}\text{'}$, we shall have $y\text{'}\; -\; y\text{'}\text{'}\; =\; a(x\text{'}\; -\; x\text{'}\text{'})$, or $a$ is determined,

9. To express the polar equation to the straight line. Let O be the pole, OY perpendicular to PR; and suppose OP joined. Let $OP\; =\; r$, $OY\; =\; p$, $\backslash angle\; POR\; =\; \backslash theta$, $\backslash angle\; YOR\; =\; \backslash alpha$; then $OP\; \backslash cos\; \backslash angle\; POY\; =\; OY$, or $r\; \backslash cos\; (\backslash theta\; -\; \backslash alpha)\; =\; p$ is the equation required.

10. b. The Circle.—The property of the circle is, that the lines drawn from the centre to the circumference are equal.

Let us suppose the axes to be rectangular, Q the centre, $QP$ the radius $=\; a$; $OM\; =\; x$, $PM\; =\; y$, the co-ordinates of P; $ON\; =\; b$, $NQ\; =\; c$, the co-ordinates of the centre; then $PR^2\; +\; QR^2\; =\; PQ^2$; or $(y-c)^2\; +\; (x-b)^2\; =\; a^2$ is the equation required. If the origin be at the centre, the equation is $x^2\; +\; y^2\; =\; a^2$.

11. The polar equation in its simplest form, when the centre is the pole, is evidently $r\; =\; a$.

12. c. The Parabola.—The property (CONIC SECTIONS, Part 1, Def. 1) is, that the distance of any point from the focus is equal to its distance from the directrix.

Let F be the focus, DQ the directrix, then $FP\; =\; PQ$. Suppose the vertex A to be the origin, and AFX the axis of the parabola to be the axis of x.

Let $AM\; =\; x$, $MP\; =\; y$, $AF\; =\; AD\; =\; a$, then $PM^2\; +\; MF^2\; =\; PQ^2\; =\; PQ^2\; =\; MD^2$; or $y^2\; +\; (x-a)^2\; =\; (x+a)^2$, which gives $y^2\; =\; 4ax$, the equation required.

13. To find the polar equation, the focus being the pole.

Let $FP\; =\; r$, $DFP\; =\; \backslash theta$; then $FP\; \backslash cos\; \backslash angle\; PFM\; =\; FM\; =\; DM\; -\; DF\; =\; PQ\; -\; DF\; =\; FP\; -\; DF\; =\; r\; -\; 2a$

14. d. The Ellipse.—The property (CONIC SECTIONS, Part 2, Def. 1) is, that the sum of the two lines drawn from any point to the two foci is constant.

Let the centre C be the origin; CA which passes through the focus S, called the semi-axis major, or the semi-transverse axis, the axis of x; CB perpendicular to CA, the semi-axis minor, or the semi-conjugate axis, the axis of y; $CM\; =\; x$, $MP\; =\; y$, $CA\; =\; a$, $CB\; =\; b$; then it is evident that $CF^2\; =\; CS^2\; =\; a^2\; -\; b^2$; and the condition $FP\; +\; SP\; =\; 2a$ gives

15. Con.—Since $a\; \backslash cdot\; SP\; =\; a^2\; -\; CS\; \backslash cdot\; x$, if $CS\; =\; ae$, we shall have $SP\; =\; a\; -\; ex$; and since $SP\; +\; FP\; =\; 2a$, $FP\; =\; a\; +\; ex$. Also it is evident that $a^2e^2\; =\; a^2\; -\; b^2$, or $e^2\; =\; 1\; -\; \backslash frac\{b^2\}\{a^2\}$. $e$ is called the eccentricity.

16. To find the polar equation to the ellipse, the focus being the pole.

Let $SP\; =\; r$, $PSA\; =\; \backslash theta$, then $PSD\; =\; \backslash pi\; -\; \backslash theta$. But (Art. 15) $SP\; =\; a\; -\; ex$,

is the equation required.

17. e. The Hyperbola.—The property is (CONIC SECTIONS, Part 3, Def. 1), that the difference of the two lines drawn from any point to the two foci is constant.

The figure being drawn, and the same letters retained as for the ellipse, we shall have $FP\; -\; SP\; =\; 2a$, which equation being treated as the corresponding equation for the ellipse, will give, first, $a\; \backslash cdot\; SP\; =\; CS\; \backslash cdot\; x\; -\; a^2$; and, finally (if $CS^2\; =\; a^2\; +\; b^2$), $a^2y^2\; -\; b^2x^2\; =\; -a^2b^2$ is the equation required.

19. f. The Cissoid of Diocles.—This curve is generated by the following construction: AQB is a semi-circle, of which AB is the diameter; AQ is any chord; QN perpendicular to the diameter; PM is drawn parallel to QN at such a distance that $AM\; =\; BN$: the point P, in which the chord AQ and the line

PM intersect one another, is a point in the cissoid APD: the cissoid is said to be the locus of the point P.

20. To find the polar equation.

Join BQ; let $AP\; =\; r$, $\backslash angle\; PAM\; =\; \backslash theta$; then because AQB is a right angle,

21. g. The Conchoid of Nicomedes.—This curve is generated as follows:—

A line of indefinite length revolves on and also slides in a fixed pivot; whilst a constant portion of the line always projects beyond a given fixed straight line, the extremity of the projecting line traces out a conchoid; or the conchoid is the locus of the extremity.

Let C be the pivot, AB the fixed line; PQ the constant portion of the revolving line which projects above it. Draw CA perpendicular to AB. Let $PQ\; =\; a$, $CA\; =\; b$, $AM\; =\; x$, $MP\; =\; y$.

By similar triangles,—
$PL^2\; :\; LC^2\; ::\; PM^2\; :\; MQ^2\; ::\; PM^2\; :\; PQ^2\; -\; PM^2$; or $(y\; +\; b)^2\; :\; x^2\; ::\; y^2\; :\; a^2\; -\; y^2$; or $x^2y^2\; =\; (y\; +\; b)^2\; (a^2\; -\; y^2)$; which is the equation required.

22. To find the polar equation. Let C be the pole,

Analytical Geometry. $CP\; =\; r$, $\backslash angle\; ACP\; =\; \backslash theta$; then since $CQ\; \backslash cos\; \backslash theta\; =\; CA$, we get $(r-a)\; \backslash cos\; \backslash theta\; =\; b$, the polar equation.

23. h. The Cycloid.—This curve is described by a point in the circumference of a circle which rolls along a straight line.

Let $GPD$ be the circle which rolls along the straight line $BR$, the point $P$ in the circumference of the circle will trace out a curve $BAR$, which is the cycloid.

Let $A$ be the highest point of the curve; $GD\; =\; 2a$ the diameter of the generating circle; $AM\; =\; x$, $MP\; =\; y$; $O$ the centre of the circle; $\backslash angle\; POG\; =\; \backslash theta$; then it is evident that the circumference of the circle coincides by succession with the straight line $BR$; therefore are $PD\; =\; BD$, arc $GPD\; =\; BC$, and arc $GP\; =\; DC$; hence

or $y\; =\; a\; \backslash text\{\; vers\}^\{-1\}\; \backslash frac\{x\}\{a\}\; +\; \backslash sqrt\{2ax\; -\; x^2\}$ is the equation required.

24. i. The Lemniscate.—The property of this curve is that the product of the two lines drawn from any point of it to the two foci is always equal to the square of half the distance between the foci.

Let $F$ and $S$ be the foci, $FC\; =\; CS\; =\; a$, $CM\; =\; x$, $MP\; =\; y$; the property is that $FP\; \backslash times\; SP\; =\; a^2$.

25. To find the polar equation. Let $CP\; =\; r$, $\backslash angle\; PCS\; =\; \backslash theta$; then

whence $r^2\; =\; 4a^2\; \backslash cos^2\; \backslash theta\; -\; 2a^2$
$=\; 2a^2(2\; \backslash cos^2\; \backslash theta\; -\; 1)$
$=\; 2a^2\; \backslash cos\; 2\backslash theta$ is the equation required.

26. Besides the cycloid there are other curves which are generated by a point in a rolling circle. The trochoid differs from the cycloid in having the generating point within the circle at a distance $b$ from the centre. Its equations are

The epicycloid is traced out by a point in the circumference of a circle which rolls on the convex circumference of another circle.

The hypocycloid is traced out by a point in the circumference of a circle which rolls on the concave circumference of a larger circle.

If the describing point is not in the circumference of the rolling circle, the curves become the epitrochoid and hypotrochoid respectively.

SECTION II.—THE DETERMINATION OF THE FORMS AND PROPERTIES OF CURVES FROM THEIR EQUATIONS.

27. Since very few curves can be described by a continuous motion, it is evident that, with rare exceptions, the determination of the form of a curve must resolve itself either into the expression of the numerical values of all coordinates corresponding to given abscissas, or into a general investigation of the number and nature of the branches of the curve, their flexure and mutual intersections. It is under the latter aspect that we are about to regard this subject.

With respect to those few curves which can be described by a continuous motion, the modes of description of some of the most important have been already given in the treatise on CONIC SECTIONS, whilst others, such as the cycloid, involve a mechanical process so obvious that it is quite superfluous to waste words in attempting to make it plain. We proceed then at once to describe the general features of a few curves from their equations. This description we shall designate as TRACING THE CURVE.

28. a. The cubical parabola of which the equation is $a^2y\; =\; x^3$.

When $x=0$, $y=0$, which shows that the curve passes through the origin. When $x$ is positive, $y$ is positive, which shows that to the right the curve lies above the axis of $x$. When $x$ is negative, $y$ is negative, which shows that to the left the curve lies below the axis of $x$.

Since $a^2\; \backslash frac\{dy\}\{dx\}\; =\; 3x^2$, it follows that

at the origin the axis of $x$ is a tangent to the curve (FLUXIONS, Art. 65, Equation 3). The form of the curve is therefore as in the figure.

29. b. The semi-cubical parabola, of which the equation is $ay^2\; =\; x^3$.

When $x=0$, $y=0$. When $x$ is positive $y^2$ is positive, or $y$ has two equal values, one + the other -; which shows that there are two similar branches of the curve to the right of the origin, the one above and the other below the axis of $x$. When $x$ is negative $y^2$ is negative, and $y$ is impossible; or there is no portion of the curve to the left of the axis of $x$. The figure is like the outline of the head of a spear, or is a cusp.

30. c. The curve whose equation is $y\; =\; \backslash frac\{x^3\}\{x^2+1\}$. It is evident that its general form will be something like that of the cubical parabola already traced. But the equation may be thrown under the form $y\; =\; x\; -\; \backslash frac\{1\}\{x\}\; +\; c$, which shows that

as $x$ increases the value of $y$ tends continually to be equal to it; which is equivalent to the fact that the curve constantly approaches a straight line whose equation is $y\; =\; x$. This straight line is called an asymptote to the curve. An example of a similar line has already been given in the article CONIC SECTIONS (iii. 15) in the case of the hyperbola. A more instructive example will be seen in the conchoid already described, where, from the nature of the case, it is at once evident that the curve continually approaches the straight line $AB$, without ever reaching it.

To ascertain whether a curve has an asymptote or not, it is sufficient to expand its equation in a descending series of powers of $x$.

If the portion which remains, after excluding that which becomes 0 on $x$ becoming infinite, be in the form of a simple equation, that equation represents the asymptote. In addition to this there may be an asymptote parallel to the axis of $y$, which can be ascertained by $y$ becoming infinite for some finite value of $x$.

31. d. There is another circumstance to be noted respecting the curve we have been discussing. Near the origin where the axis of $x$ is a tangent, it is evidently convex to that axis; but from the nature of the case it is clear that at a considerable distance from the origin it must be convex towards its asymptote, and consequently concave towards the axis of $x$. The point at which it changes from convex to concave is called a point of inflexion or a point of contrary flexure. To ascertain the position of this point, it is necessary to find the value of $x$ which shall render $\backslash frac\{d^2y\}\{dx^2\}\; =\; 0$.

Analytical Geometry. On solving this equation we shall obtain $x\; =\; \backslash pm\; \backslash sqrt\{3\}$, which gives the point of inflexion. The curve is therefore as follows (fig. 11), OB being the asymptote, and P the point of inflexion.

32. e. To trace the curve of which the equation is

Suppose $a$ the least, and $c$ the greatest of the three magnitudes. We have

from which it is clear that in all cases there are two values of $y$ equal and with opposite signs; or, in other words, there is symmetry with respect to the axis of $x$.

When $x=0$, $y$ is infinite, or the axis of $y$ is an asymptote.

When , $y$ is impossible, or no branch of the curve lies between the origin and the distance $a$ to the right.

When $x=a$, $y=0$; when , $y$ is possible; and when $x=b$, $y=0$; the curve accordingly starts from the axis of $x$ at the distance $a$ from the origin, and returns to it again at the distance $b$. This branch of the curve is consequently a kind of oval. If it be required to ascertain where is the broadest part of the oval, it will be necessary to find at what point $y$ is a maximum by the method given in FLUXIONS, art. 60. The result is obtained by the solution of a cubic equation.

When , $y$ is impossible.

When $x=c$, $y=0$; when $x\; >\; c$, $y$ is possible; and when $x$ is infinite, $y$ is infinite—the branch beyond the distance $c$ consequently starts from the axis of $x$, and extends to infinity, somewhat in the form of a parabola. When $x$ is negative, $y$ is always possible; and it is infinite both when $x$ is 0 and when $x$ is infinite—diminishing from the first point and then increasing to the second. At some point it must attain a minimum value, which can be ascertained as the maximum above. The branch to the left of the origin accordingly bears some faint resemblance to an hyperbola. The whole curve is as in the figure where O is the origin—$OA=a$, $OB=b$, $OC=c$.

33. f. In the last example we assumed that $a$, $b$, and $c$ are all different from each other. If we suppose $a=b$, we have

the equation $y\; =\; (x-a)\; \backslash sqrt\{\backslash frac\{x-c\}\{x\}\}$. The peculiarity in this equation is this—that when $x=a$, $y=0$, which gives a possible point A; but when $x$ is a little greater or less than $a$, $y$ is impossible, so that no branch of the curve lies near the point. This point, an isolated point where co-ordinates satisfy the equation, but which does not belong to any branch of the curve, is called a conjugate point. By comparing this example with the last, of which it is a particular case, we shall be able to trace the origin of such a point in the oval AB, which has collapsed.

We will now give an example or two of the use of polar co-ordinates.

34. g. The Lemniscate, of which the equation (i) is $r^2\; =\; 2a^2\; \backslash cos\; 2\backslash theta$.

When $\backslash theta=0$, $r^2=2a^2$, $r=\backslash pm\; a\backslash sqrt\{2\}$; or $r$ has two equal values, CA and CB in the figure of Art. 24.

Analytical Geometry. When , $r$ is possible, having two equal values, positive and negative, the former giving the upper half of the right-hand portion, and the latter the lower half of the left.

When $\backslash theta\; =\; \backslash frac\{\backslash pi\}\{4\}$, $r=0$; when , $r$ is impossible.

When $\backslash theta\; =\; \backslash frac\{3\backslash pi\}\{4\}$, $r=0$; when $\backslash theta\; =\; \backslash pi$, $r=\backslash pm\; a\backslash sqrt\{2\}$, thus giving the upper portion of the left-hand figure, and the lower portion of the right.

35. h. The Cardioide, of which the equation is

When $\backslash theta=0$, $r=a$; when $\backslash theta$ is , $r$ is positive; when $\backslash theta\; =\; \backslash pi$, $r=0$; and the same is true in a reverse order between $\backslash pi$ and $2\backslash pi$. The curve, therefore, starts at its greatest distance from the origin, and returns to the origin by having positive radii throughout a whole circumference—a portion of it accordingly lies to the left of the axis of $y$. The curve is in the shape of a heart, thus: O being the origin; $OA=a$.

Many curves, whose equations are presented between linear co-ordinates, have their general form, and some of their properties, determined most readily by converting the given equation into a polar equation, as in the following example:—

Let $x=r\; \backslash cos\; \backslash theta$, $y=r\; \backslash sin\; \backslash theta$; then the equation gives $r\; =\; a\; \backslash frac\{\backslash sin\; \backslash theta\}\{\backslash cos^3\; \backslash theta\}\; (\backslash cos^2\; \backslash theta\; -\; \backslash sin^2\; \backslash theta)$.

When $\backslash theta=0$, $r=0$; when , $r$ is positive; when $\backslash theta\; =\; \backslash frac\{\backslash pi\}\{4\}$, $r=0$; hence there is an oval in the first octant, having its extremity at the origin.

When , $r$ is negative; when $\backslash theta\; =\; \backslash frac\{\backslash pi\}\{2\}$, $r$ is infinite; hence there is no branch of the curve in the second octant, but one extending back from the origin to infinity in the sixth octant.

When , $r$ is negative; when $\backslash theta\; =\; \backslash frac\{3\backslash pi\}\{4\}$, $r=0$; hence there is no branch of the curve in the third octant, but a branch returning from infinity to the origin in the seventh.

When , $r$ is positive; when $\backslash theta\; =\; \backslash pi$, $r=0$; hence there is an oval in the fourth octant, having its extremity at the origin. The maximum radius vector occurs at about 30. The curve is therefore in the annexed form.

37. These examples will suffice to illustrate the general determination of the form of a curve from its equation. We have next to discuss the determination of the properties of curves from their equations. This we shall do very briefly, confining ourselves to curves of the second degree. As a preliminary it will be requisite to point out the method employed to transfer our reference from one system of co-ordinates to another. This method is called

38. It is supposed that the equation to a curve referred to a certain system of co-ordinates $x,\; y$, is given; and it is required to determine what its equation becomes when referred to another system of co-ordinates $x\text{'},\; y\text{'}$. The problem obviously resolves itself into the determination of the values of $x$ and $y$ in terms of $x\text{'}$ and $y\text{'}$, and the substitution of the latter in place of the former. The formulae are thus obtained.

39. 1°. If the new axes are parallel to the old, the origin alone being changed—it is evident that nothing is required but to write $x\text{'}\; +\; a$ in place of $x$, and $y\text{'}\; +\; b$ in place of $y$; where $a$ and $b$ are the co-ordinates of the new origin referred to the old.

40. 2°. Suppose the origin unchanged, and both the old and new axes rectangular, the axis of $x\text{'}$ being inclined to that of $x$ by the angle $\backslash theta$. Let $OM\; =\; x$, $MP\; =\; y$, $ON\; =\; x\text{'}$, $NP\; =\; y\text{'}$. Draw $NR,\; NQ$ parallel to $Oy$ and $Ox$ respectively;

then $x\; =\; OR\; -\; QN\; =\; x\text{'}\; \backslash cos\; \backslash theta\; -\; y\text{'}\; \backslash sin\; \backslash theta$,
$y\; =\; NR\; +\; PQ\; =\; x\text{'}\; \backslash sin\; \backslash theta\; +\; y\text{'}\; \backslash cos\; \backslash theta$.

41. 3°. Suppose the origin to be unchanged; and neither the old nor the new axes to be necessarily rectangular.

Let $OM\; =\; x$, $MP\; =\; y$, $ON\; =\; x\text{'}$, $NP\; =\; y\text{'}$. Draw $ME,\; NF$ perpendicular to $Oy$, and let $NG$ be parallel to $Oy$ and meet $EM$ produced in $G$; then $ME\; =\; NF\; -\; GM$, or $x\; \backslash sin\; x\text{'}Oy\; =\; x\text{'}\; \backslash sin\; x\text{'}Oy\; -\; y\text{'}\; \backslash sin\; y\text{'}Oy$. Similarly, by drawing perpendiculars from $P$ and $N$ on $Ox$, we get

42. 4°. When both the origin and the direction of the axes are to be changed, we have only to unite the processes of No. 1 and Nos. 2 and 3 to effect the transformation.

43. 5°. To obtain a polar equation, let $OP\; =\; r$;

Properties of Curves of the Second Degree, as deduced from the general equation.

The general equation of the second degree is
$Ay^2\; +\; 2Bxy\; +\; Cx^2\; +\; 2Dy\; +\; 2Ex\; +\; F\; =\; 0$.

44. PROP. I.—No straight line can cut a curve of the second degree in more than two points.

Let $y\; =\; mx\; +\; n$ be the equation to a straight line, then by substituting this value of $y$ in the given equation, we have a quadratic for the determination of the values of $x$ at the points where the co-ordinates are common to it and to the straight line. A similar process gives a quadratic equation in $y$. Consequently there are but two solutions, and therefore but two points of intersection.

By referring to the equations to the conic sections given above, it will be seen that they are curves of the second degree.

45. PROP. II.—Admitting that the above equation represents a conic section, it is required to determine the particular one.

If we solve the equation with respect to $y$, we obtain

Now, I. If $B^2\; -\; AC$ be negative, this value of $y$ is impossible when $x$ is infinite; which shows that the curve does not extend to an infinite distance. It is therefore a circle or an ellipse, according as $C$ is equal to $A$ or not. The ellipse, however, may be simply a point, and the equation may be impossible.

II. If $B^2\; -\; AC$ be positive, $y$ has two real roots when $x$ is infinite, whether $x$ be positive or negative. The curve has consequently four infinite branches as in the hyperbola or in two intersecting straight lines.

III. If $B^2\; -\; AC\; =\; 0$; then if $BD\; -\; AE$ is positive, $y$ will be possible when $x$ is positive and infinite, but impossible when $x$ is negative, and vice versa. The curve is consequently a parabola. If $BD\; -\; AE\; =\; 0$, the equation represents two straight lines or one, or is impossible.

IV. If the quantity under the radical is a complete square, i. e., if $(B^2\; -\; AC)(D^2\; -\; AF)\; =\; (BD\; -\; AE)^2$ the equation represents two straight lines; for it is the product of two simple equations.

46. PROP. III.—To find the centre of the curve.

The centre is that point which bisects every chord which can be drawn through it. Let $a,\; b$ be the co-ordinates of the centre; $x\text{'},\; y\text{'}$ the co-ordinates when the centre is the origin; then (Art. 39) we have $x\; =\; x\text{'}\; +\; a$, $y\; =\; y\text{'}\; +\; b$; whence $A(y\text{'}\; +\; b)^2\; +\; 2B(x\text{'}\; +\; a)(y\text{'}\; +\; b)\; +\; C(x\text{'}\; +\; a)^2\; +\; 2D(y\text{'}\; +\; b)\; +\; 2E(x\text{'}\; +\; a)\; +\; F\; =\; 0$.

Now when $y\text{'}\; =\; 0$, this equation must give two values of $x\text{'}$, equal and with opposite signs, because the chord of $x\text{'}$ is bisected at the centre. Hence their sum is equal to 0;

which are the co-ordinates of the centre.

In the case of the parabola these co-ordinates are infinite, for (Art. 45) $B^2\; -\; AC\; =\; 0$; hence we may say that the parabola has no centre. In analysis, however, it is desirable to regard the centre as existing at an infinite distance.

47. PROP. IV.—To find the locus of the middle points of parallel chords.

Let $a,\; b$ be the co-ordinates of one of those middle points, and let the origin be transferred to that point by writing $x\text{'}\; +\; a,\; y\text{'}\; +\; b$ for $x$ and $y$. Let also $y\text{'}\; =\; mx\text{'}$ be the equation to this chord; then the points in which it intersects the curve will be those for which $x\text{'}$ and $y\text{'}$ are the same in the equation to the curve and to the chord. Hence

is the equation for determining the values of $x\text{'}$ at the points of section; but since these are by hypothesis equal and with opposite signs, the co-efficient of $x\text{'}$ must be equal to 0;

Now $m$ remains the same for all chords which are parallel. This equation is therefore a simple equation between $a$ and $b$; consequently if $a$ and $b$ be considered as the co-ordinates of middle points generally, the equation which connects them is the equation to a straight line. The locus of the middle points of all parallel chords is therefore a straight line.

Analytical Geometry. 48. Every straight line which bisects a system of parallel chords is called a diameter, and the chords themselves are called ordinates to that diameter.

49. COR.—From the definition of a centre it is evident that every diameter passes through it.

In the case of the parabola, which has no finitely situated centre, we observe that $B^2\; =\; AC$. Now the equation to the diameter (Art. 47) is

in which the co-efficients of the co-ordinates $a$ and $b$ are independent of $m$, which shows (Art. 8) that all diameters of the parabola are parallel to one another.

50. PROP. V.—If two diameters of a curve of the second degree be such that one of them bisects all chords parallel to the other, then the latter also will bisect all chords parallel to the former.

By Art. 47 the equation to the line which bisects all chords parallel to the line whose equation is $y\; =\; mx$, is

by writing $y$ and $x$ for the symbols which stand for the general phrases latitude and longitude of a point in the line. Let this be abbreviated by $y\; =\; m\text{'}x\; +\; \backslash dots$,

In the same manner the diameter which bisects all chords parallel to $y\; =\; m\text{'}x$ is $y\; =\; m\text{'}\text{'}x$;

i.e. (Art. 8), this diameter is parallel to the original chord; hence the truth of the proposition. Diameters so related to each other are called conjugate diameters.

51. PROP. VI.—If through any point $O$ two chords be drawn, meeting the curve in the points $P,\; Q$, and $R,\; S$, respectively; then, the ratio of the rectangle $OP\; \backslash cdot\; OQ$, to the rectangle $OR\; \backslash cdot\; OS$, is a ratio which is the same whatever be the point $O$, provided the direction of the chords remain unchanged.

Let the co-ordinates of $O$ be $a$ and $b$; the polar radius through $O\; =\; r$; then (Art. 43) we have

where $m$ and $n$ depend on the direction of the radius or chord.

The general equation becomes, by substitution, $A(nr\; +\; b)^2\; +\; 2B(mr\; +\; a)(nr\; +\; b)\; +\; C(mr\; +\; a)^2\; +\; 2D(nr\; +\; b)\; +\; 2E(mr\; +\; a)\; +\; F\; =\; 0$; which gives two values of $r$, viz. $OP,\; OQ$. Their product is the last term of the equation, viz.

similarly the product $OR\; \backslash cdot\; OS$ is

Consequently $OP\; \backslash cdot\; OQ\; :\; OR\; \backslash cdot\; OS\; ::\; An^2\; +\; 2Bmn\; +\; Cm^2\; :\; An^2\; +\; 2Bmn\; +\; Cm^2$;

a ratio which is independent of $a$ and $b$, and therefore of the position of the point $O$. If $O\text{'}$ be any other point, and $O\text{'}P\text{'}$ be parallel to $OP,\; O\text{'}R\text{'}$ to $OR$, &c.; then $OP\; \backslash cdot\; OQ\; :\; OR\; \backslash cdot\; OS\; ::\; O\text{'}P\text{'}\; \backslash cdot\; O\text{'}Q\text{'}\; :\; O\text{'}R\text{'}\; \backslash cdot\; O\text{'}S$.

52. COR. 1.—If $O\text{'}$ be the centre; $O\text{'}P\text{'}\; =\; O\text{'}Q\text{'},\; O\text{'}R\text{'}\; =\; O\text{'}S$; then $OP\; \backslash cdot\; OQ\; :\; OR\; \backslash cdot\; OS\; ::\; O\text{'}P\text{'}^2\; :\; O\text{'}R\text{'}^2$. That is, if two chords intersect one another, the rectangles by their segments are to one another as the squares of the diameters parallel to them respectively.

53. COR. 2.—$O\text{'}$ being still the centre, let $O\text{'}O$ be the diameter of which $PQ$ is the ordinate; and let $RS$ and also $R\text{'}S\text{'}$ pass through $O\text{'}$ and coincide with one another; then $OP^2\; :\; OR\; \backslash cdot\; OS\; ::\; O\text{'}P\text{'}^2\; :\; O\text{'}R\text{'}^2$; or the square of the ordinate is to the rectangle by the abscissas, as the square of the diameter which is parallel to the former to the square of that which passes through the latter.

54. In the case of the parabola this proposition is inapplicable; for the diameter and the chord parallel to it are both infinite. To determine the corresponding ratio for this case, let the axes be transformed to $x\text{'},\; y\text{'}$ such that the axis of $x\text{'}$ is parallel to the diameter, and the axis of $y\text{'}$ to a bisected chord. Let also the origin be on the curve—the equation will then assume the form $Ay\text{'}^2\; +\; 2Ex\text{'}\; =\; 0$; for $C\; =\; 0$; because when $y\; =\; 0$ there must be only one value of $x\text{'}$, viz. $x\text{'}\; =\; 0$; but $B^2\; =\; 4AC$, $\backslash therefore\; B\; =\; 0$; and since the two values of $y$ are equal and with opposite signs we must have $D\; =\; 0$; lastly, since the origin is on the curve we must have $F\; =\; 0$. The equation in this form shows that the abscissas are to one another as the squares of the ordinates.

The theorems contained in this second corollary constitute the fundamental properties of the conic sections, and are the same as demonstrated in the treatise on that subject. (Arts. i. 12, ii. 13, iii. 12.)

55. PROP. VII.—To find the equation to the tangent.

The tangent is that line to which a line cutting the curve in two points continually approaches as its limit, as the points of section approach to each other.

The equation to a line which meets the curve in the two points $x\text{'}y\text{'},\; x\text{'}\text{'}y\text{'}\text{'}$ is

Now $Ay\text{'}^2\; +\; 2Bx\text{'}y\text{'}\; +\; Cx\text{'}^2\; +\; 2Dy\text{'}\; +\; 2Ex\text{'}\; +\; F\; =\; 0\; \backslash dots\; \backslash dots\; \backslash dots\; (1)$ and $Ay\text{'}\text{'}^2\; +\; 2Bx\text{'}\text{'}y\text{'}\text{'}\; +\; Cx\text{'}\text{'}^2\; +\; 2Dy\text{'}\text{'}\; +\; 2Ex\text{'}\text{'}\; +\; F\; =\; 0$;

by subtraction,

and the equation to the cutting line is

The equation to the tangent is derived from this by writing $y\text{'}$ for $y\text{'}\text{'}$ and $x\text{'}$ for $x\text{'}\text{'}$. It is therefore

Those who are familiar with the differential calculus will obtain this result at once from the equation

If we multiply out equation (2), and substitute for $Ay\text{'}^2\; +$

Analytical Geometry. $2Bx\text{'}y\text{'}\; +\; Cx\text{'}^2\; +\; Dy\text{'}\; +\; Ex\text{'}$ its value $-Dy\text{'}\; -\; Ex\text{'}\; -\; F$ from equation (1), we shall have

$Axy\text{'}\; +\; B(xy\text{'}\; +\; x\text{'}y)\; +\; Cxx\text{'}\; +\; D(y\; +\; y\text{'})\; +\; E(x\; +\; x\text{'})\; +\; F\; =\; 0$ as the equation to the tangent. The equation to the tangent is therefore to be got from the equation to the curve by writing $xx\text{'}$ for $x^2$, $yy\text{'}$ for $y^2$, $x\; +\; x\text{'}$ for $2x$, $y\; +\; y\text{'}$ for $2y$, and $xy\; +\; xy\text{'}$ for $2xy$; or by changing one of the $x$'s into $x\text{'}$, and one of the $y$'s into $y\text{'}$.

56. COR.—Since the equation to the tangent is symmetrical in $x$ and $x\text{'}$, $y$ and $y\text{'}$, if we abbreviate it by $\backslash phi(x,\; y,\; x\text{'},\; y\text{'})$, we shall have the equation $\backslash phi(x,\; y,\; x\text{'},\; y\text{'})\; =\; \backslash phi(x\text{'},\; y\text{'},\; x,\; y)$; $x$ and $y$ being interchangeable with $x\text{'}$ and $y\text{'}$.

57. PROP. VIII.—To find the equation to the straight line which joins the points of contact of two tangents, real or imaginary.

Abbreviate the equation to the curve by $f(x,\; y)\; =\; 0$, and to the tangent by $\backslash phi(x,\; y,\; x\text{'},\; y\text{'})\; =\; 0$.

Let $X,\; Y$ be the co-ordinates of the point of intersection of two tangents; then the equations $f(x\text{'},\; y\text{'})\; =\; 0$, $\backslash phi(x\text{'},\; y\text{'},\; X,\; Y)\; =\; 0$ give the values of $x\text{'},\; y\text{'}$, the co-ordinates of the point of contact of the one tangent.

Also the equations $f(x\text{'}\text{'},\; y\text{'}\text{'})\; =\; 0$, $\backslash phi(x\text{'}\text{'},\; y\text{'}\text{'},\; X,\; Y)\; =\; 0$, give the co-ordinates $x\text{'}\text{'},\; y\text{'}\text{'}$ of the point of contact of the other tangent which passes through the point $X,\; Y$.

Now, of the equations constituting these respective pairs, one is a simple equation, and the other a quadratic. Each pair will therefore give two values of the variables. But each pair is the same as the other; therefore the two roots of the one are the roots of both; or $x\text{'},\; y\text{'},\; x\text{'}\text{'},\; y\text{'}\text{'}$ are the roots of the equations $f(x,\; y)\; =\; 0$, $\backslash phi(x,\; y,\; X,\; Y)\; =\; 0$.

Again, let $\backslash psi(x,\; y,\; X,\; Y)\; =\; 0$ be the equation to the chord which passes through the points $(x\text{'},\; y\text{'});\; (x\text{'}\text{'},\; y\text{'}\text{'})$ of contact; then will $x\text{'},\; y\text{'},\; x\text{'}\text{'},\; y\text{'}\text{'}$ be the roots of the equations $f(x,\; y)\; =\; 0$, $\backslash psi(x,\; y,\; X,\; Y)\; =\; 0$.

But it has been already proved that

have the same roots. Moreover, one of the equations, $f(x,\; y)\; =\; 0$, is common to both pairs; therefore the other two equations must be identical; or $\backslash psi(x,\; y,\; X,\; Y)\; =\; 0$ is the same as $\backslash phi(x,\; y,\; X,\; Y)\; =\; 0$; i.e., the equation to the line which joins the points of contact of two tangents is of the same form as the equation to the tangent; or if the equation to the tangent at the point $x\text{'},\; y\text{'}$ in the curve is $\backslash phi(x,\; y,\; x\text{'},\; y\text{'})\; =\; 0$; the equation to the line which joins the points of contact of two tangents which meet in a point $x\text{'},\; y\text{'}$, out of the curve is $\backslash phi(x,\; y,\; x\text{'},\; y\text{'})\; =\; 0$.

58. The line which joins the points of contact of two tangents is called the polar of their point of intersection; and that point accordingly is called the pole of the line.

59. PROP. IX.—If the point $B$ is in the polar of $A$, then is $A$ also in the polar of $B$.

Let $x\text{'},\; y\text{'}$ be the co-ordinates of $A$,
$x\text{'}\text{'},\; y\text{'}\text{'}$ ... $B$.

The equation to the polar of $A$ is $\backslash phi(x,\; y,\; x\text{'},\; y\text{'})\; =\; 0$; but by hypothesis $B(x\text{'}\text{'},\; y\text{'}\text{'})$ is a point in this line; therefore $\backslash phi(x\text{'}\text{'},\; y\text{'}\text{'},\; x\text{'},\; y\text{'})\; =\; 0$. Hence also (Art. 56) $\backslash phi(x\text{'},\; y\text{'},\; x\text{'}\text{'},\; y\text{'}\text{'})\; =\; 0$. But $\backslash phi(x,\; y,\; x\text{'},\; y\text{'})\; =\; 0$ is the equation to the polar of $B$; the equation $\backslash phi(x\text{'},\; y\text{'},\; x\text{'}\text{'},\; y\text{'}\text{'})\; =\; 0$ consequently shows that $x\text{'},\; y\text{'}$ are the co-ordinates of a point in that line; i.e., $A$ is a point in the polar of $B$.

60. COR. 1.—If any number of points be taken in the polar of $A$, their polars will

all pass through $A$; i.e., the polars of every point in a straight line all pass through the pole of that line.

61. COR. 2.—If $A$ and $B$ are the poles of two lines $PQ$ and $RS$ which meet in $O$, the line $AB$ is the polar of the point $O$.

For the polar of $O$ passes through $A$ (by the Proposition) and through $B$; it is therefore the straight line $AB$.

62. PROP. X.—To find the condition that the straight line whose equation is $my\; +\; nx\; +\; p\; =\; 0$ may touch a given conic section.

The equation to the conic section may, by properly selecting the co-ordinates, be thrown into one or other of the two forms,

The tangents to these have for their respective equations (Art. 55),

In order that the former of these may coincide with the equation $my\; +\; nx\; +\; p\; =\; 0$, we must have

But $\backslash frac\{x\text{'}^2\}\{a\}\; +\; \backslash frac\{y\text{'}^2\}\{b\}\; =\; 1$; whence $a\; n^2\; +\; b\; m^2\; =\; p^2$.....(1)

is the resulting condition.

In order that the latter of the above equations may coincide with the equation $my\; +\; nx\; +\; p\; =\; 0$, we must have

which being substituted in the equation

as the condition required.

63. PROP. XI.—Given that chords to a curve of the second degree always touch a conic section, it is required to prove that the locus of their poles is a conic section.

Let $A$ be the conic section to which the chords are always tangents; $B$ the curve of the second degree (a conic section) to which the chords are drawn; $C$ the curve which is the locus of the poles of the chords to $B$.

Let $x\text{'},\; y\text{'}$ be the co-ordinates of a point in $C$, the equation to the polar of this point is

In order that this line may be a tangent to the conic section $A$, whose equation we shall suppose to be either

we have to satisfy one or other of the two conditions obtained in the last proposition, viz.

Analytical Geometry. or $a(By\text{'}\; +\; Cx\text{'}\; +\; E)^2\; +\; b(Ay\text{'}\; +\; Bx\text{'}\; +\; D)^2\; =\; (Dy\text{'}\; +\; Ex\text{'}\; +\; F)^2$ (1)
or $a^2(By\text{'}\; +\; Cx\text{'}\; +\; E)^2\; +\; 2a(By\text{'}\; +\; Cx\text{'}\; +\; E)(Dy\text{'}\; +\; Ex\text{'}\; +\; F)\; +\; b(Ay\text{'}\; +\; Bx\text{'}\; +\; D)^2\; =\; 0$ ..... (2)
as necessary to be fulfilled.

If the requisite one of these relations between $x\text{'}$ and $y\text{'}$ hold true, the polar to every point in $C$ will touch $A$. But these conditions are both expressed in the form of a quadratic equation between $x\text{'}$ and $y\text{'}$. Hence the curve $C$ is a curve of the second degree or a conic section.

64. PROP. XII.—Given that $C$ is generated by $A$ through the intervention of $B$, as in the last proposition, to prove that $A$ would be generated by $C$ through the intervention of the same curve $B$.

Let two tangents be drawn to $A$, forming chords to $B$, and let the poles of those chords be $C$ and $C\text{'}$ in the curve $C$: then (Art. 61) the point of intersection of those tangents is the pole of $CC\text{'}$ (a chord in $C$). But as the tangents become indefinitely near each other, their point of intersection tends to become the point $A$ in the curve $A$; and the points $C,\; C\text{'}$ also tend to become indefinitely near each other, or the chord $CC\text{'}$ tends to become a tangent. Hence the point $A$ on the curve $A$ is the pole of the tangent to a point $C$ on the curve $C$.

65. The curves $A$ and $C$ are therefore reciprocal, and the process which employs this method of demonstration is termed the method of reciprocal polars. We shall confine ourselves to a single application of this method, that of connecting two theorems relative to the conic sections which are known as PASCAL'S and BRIANCHON'S theorem respectively.

Pascal's Theorem.

66. PROP. XIII.—If any hexagon be inscribed in a conic section, and the opposite sides be produced to meet in three points, those points are in a straight line.

Let $ABCDEF$ be a hexagon inscribed in a conic section, and let the opposite sides produced meet in the

Fig. 18.

points $G,\; H$, and $K$; viz., $AB$ and $DE$ in $G$; $BC$ and $EF$ in $H$; $CD$ and $FA$ in $K$: then the points $G,\; H$, and $K$

are in a straight line. Let $u=0,\; v=0,\; w=0$, be three simple equations between $x$ and $y$, each multiplied by an arbitrary constant, as in Art. 7; and let $u=0$ (1) be the equation to the line $AB,\; v=0$ (2) to the line $CD$, and $w=0$ (3) to the line $EF$.

Now since $u,\; v,\; w$, involve $x$ and $y,\; u^2,\; v^2,\; w^2$, &c., will involve $x^2,\; y^2$, &c., consequently

is an equation between $x,\; y,\; x^2,\; y^2$, &c., containing six arbitrary constants; and is therefore the general equation of the second degree which represents any conic section.

At the points $A$ and $B$, where the straight line represented by equation (1) intersects the conic section represented by (4), the same values of $x$ and $y$ render both equations identical, or $u$ is the same in fact as well as in form in both equations:

We must have, therefore,

at those points respectively. Similarly at the points $D$ and $C$ we must have $u\; =\; \backslash mu\; w,\; u\; =\; \backslash frac\{1\}\{\backslash mu\}\; w$ respectively, and at the points $E$ and $F,\; u\; =\; \backslash nu\; w$ and $u\; =\; \backslash frac\{1\}\{\backslash nu\}\; w$. Hence the points are thus determined, viz.:

Let $u\; =\; Av\; +\; Bw$ be the equation to the line which passes through the points $B$ and $C$; then, by (6) we must have $0\; =\; \backslash frac\{A\}\{\backslash lambda\}\; +\; B$; and by (7) $\backslash frac\{1\}\{\backslash mu\}\; =\; B$; consequently

is the equation to the line $BC$. Similarly $u\; =\; \backslash nu\; v\; +\; \backslash mu\; w$ (12)

is the equation to the line $DE$, and $u\; =\; \backslash frac\{1\}\{\backslash nu\}\; (v\; -\; \backslash lambda\; w)$ (13) is

the equation to the line $FA$.

Let $Au\; +\; Bv\; +\; Cw\; =\; 0$ be the equation to $GK$.

This equation and the equations to $AB$ and $ED$ must be simultaneously satisfied at the point $G$. Combining then

(1) and (12) with this equation we get $\backslash frac\{\backslash mu\}\{C\}\; =\; \backslash frac\{\backslash nu\}\{B\}$.

In like manner the equation must be satisfied simultaneously with those to $AF$ and $CD$, at the point $K$. The equations (2) and (13) combined with it give $\backslash frac\{\backslash nu\}\{A\}\; =\; \backslash frac\{\backslash lambda\}\{C\}$, whence

$A\; =\; \backslash frac\{1\}\{\backslash lambda\},\; B\; =\; \backslash frac\{1\}\{\backslash mu\},\; C\; =\; \backslash frac\{1\}\{\backslash nu\}$; and the equation to the line $GK$ becomes

Again, let $Au\; +\; Bv\; +\; Cw\; =\; 0$ be the equation to the line $GH$. The point $G$ gives as before $\backslash frac\{\backslash mu\}\{C\}\; =\; \backslash frac\{\backslash nu\}\{B\}$; at the point $H$ the equation must be satisfied simultaneously with the equa-

Analytical Geometry. tions to BC and EF. Combining (3) and (11) with it, we get $\backslash frac\{\backslash mu\}\{A\}\; =\; \backslash frac\{\backslash lambda\}\{B\}$; whence we obtain $A\; =\; \backslash frac\{1\}\{\backslash lambda\}$, $B\; =\; \backslash frac\{1\}\{\backslash mu\}$, $C\; =\; \backslash frac\{1\}\{\backslash nu\}$; and

the equation to GH is $\backslash frac\{u\}\{\backslash lambda\}\; +\; \backslash frac\{v\}\{\backslash mu\}\; +\; \backslash frac\{w\}\{\backslash nu\}\; =\; 0$, the same as the equation to GK. Hence GH and GK are in the same straight line.

67. CON.—It is evident that the demonstration does not require any order of position amongst the points, or, in other words, that the points are interchangeable. Consequently, if for hexagon we read six points, and admit every variety of modes of joining them, we shall obtain a great number (sixty) of different straight lines in which three points of intersection of joining lines lie.

Brianchon's Theorem.

68. PROP. XIV.—If any hexagon be described about a conic section, the three diagonals which join opposite angles meet in a point.

Let two tangents be drawn to a conic section A as in Art. 63, meeting in the point P, and let their poles in the conic section C be $a,\; b$; then (Art. 61) P is the pole of the line $ab$. The same is true of any other pair of tangents; consequently, if six tangents be drawn to A forming a hexagon about it, their points of intersection will be the poles of a hexagon in C.

Let us designate the angles of the hexagon described about A by P, Q, R, S, T, U; and the sides of the hexagon described in C to which they are respectively poles by $p,\; q,\; r,\; s,\; t,\; u$.

Let $p$ and $s$ meet in G; $q$ and $t$ in H, and $r$ and $u$ in K; then, by the last proposition, G, H, and K are in a straight line.

Now (Art. 61) G is the pole of PS, H of QT, and K of RU; and since G, H, and K are in a straight line, their polars PS, QT, RU pass through the pole of that line (Art. 60); or meet in a point.

SECTION III.—DEDUCTION OF PROPERTIES OF CURVES FROM THEIR EQUATIONS AND KNOWN PROPERTIES.

We propose in this section to deduce some of the more important properties of the conic sections from their equations, combined with their known geometrical figures. The axes of co-ordinates will be supposed rectangular unless otherwise expressed.

1. The Straight Line.

69. PROP. I.—To find the equation to a straight line which shall be perpendicular to a given straight line.

Let $y\; =\; ax\; +\; b$ be the equation to the given straight line. $y\; =\; px\; +\; q$ that of the required line.

If $\backslash theta,\; \backslash varphi$ be the angles in which those lines respectively cut the axis of $x$; it is evident that the condition imposed requires that

hence $y\; =\; -\backslash frac\{1\}\{a\}x\; +\; q$ is the equation required.

70. PROP. II.—To find the length of the perpendicular from a given point on a given line.

Let $x\text{'},\; y\text{'}$ be the co-ordinates of the given point A; $y\; =\; ax\; +\; b$ the equation to the given line PR.

Draw AP perpendicular to the given line, and let $y\text{'}$ cut RP in E;

71. We may exhibit this result in a very different form by adopting the equation to a straight line given in Art. 7.

Let $AP\; =\; P$; then the equation to a line through A parallel to RP is $y\; \backslash cos\; \backslash alpha\; -\; x\; \backslash sin\; \backslash alpha\; -\; (p\; +\; P)\; =\; 0$; and since A is a point in this line,

72. PROP. III.—If $\backslash alpha\; =\; 0,\; \backslash beta\; =\; 0$ be the equations to two straight lines, agreeably to the abbreviated notation of Art. 7; then $la\; +\; m\backslash beta\; =\; 0$, or $\backslash alpha\; -\; k\backslash beta\; =\; 0$ will represent a line which passes through their point of intersection.

For the co-ordinates which render both $\backslash alpha\; =\; 0$ and $\backslash beta\; =\; 0$ render $la\; +\; m\backslash beta\; =\; 0$, or $\backslash alpha\; -\; k\backslash beta\; =\; 0$; and are therefore co-ordinates of a point in the last line as in the other two.

73. PROP. IV.—If $\backslash alpha\; =\; 0$ be the equation to the line QA, $\backslash beta\; =\; 0$ to the line QB, and $\backslash alpha\; -\; k\backslash beta\; =\; 0$ to the line QC;

Let $x\text{'},\; y\text{'}$ be the co-ordinates of C. Draw CA, CB perpendicular to QA, QB; then (Art. 71) $CA\; =\; y\text{'}\; \backslash cos\; \backslash alpha\; -\; x\text{'}\; \backslash sin\; \backslash alpha\; -\; p$.

$CB\; =\; \backslash beta$, when $x\text{'},\; y\text{'}$ are written for the co-ordinates. But since $x\text{'},\; y\text{'}$ is a point in the line $\backslash alpha\; -\; k\backslash beta\; =\; 0$, we have

74. COR. 1.—The equation to the line QC is therefore $\backslash alpha\; \backslash sin\; BQC\; -\; \backslash beta\; \backslash sin\; AQC\; =\; 0$.

75. COR. 2.—The straight line which bisects the angle between two straight lines whose equations are $\backslash alpha\; =\; 0,\; \backslash beta\; =\; 0$ is $\backslash alpha\; -\; \backslash beta\; =\; 0$.

76. PROP. V.—If $\backslash alpha\; =\; 0,\; \backslash beta\; =\; 0,\; \backslash gamma\; =\; 0$ be the equations to three straight lines, and it be possible to find three numerical quantities, $l,\; m,\; n$, such that $la\; +\; m\backslash beta\; +\; n\backslash gamma$ shall be equal to 0; the three straight lines shall meet in one point.

At the point where two of them whose equations are $\backslash alpha\; =\; 0,\; \backslash beta\; =\; 0$ intersect, the co-ordinates render $la\; +\; m\backslash beta\; =\; 0$. But, by hypothesis, $la\; +\; m\backslash beta\; +\; n\backslash gamma\; =\; 0$ for all co-ordinates;

Fig. 20.

Fig. 21.

$\backslash therefore\; \backslash pi\backslash gamma\; =\; 0$ for the co-ordinates of the point where the first and second lines meet; consequently the same co-ordinates belong to the third line; or the three meet in that point.

77. PROP. VI.—The straight lines which bisect the angles of a triangle meet in a point.

Let $\backslash alpha\; =\; 0,\; \backslash beta\; =\; 0,\; \backslash gamma\; =\; 0$ be the equations to the sides of the triangle; $\backslash alpha\; -\; \backslash beta\; =\; 0,\; \backslash alpha\; -\; \backslash gamma\; =\; 0,\; \backslash beta\; -\; \backslash gamma\; =\; 0$ (Art. 75) are the equations to the lines which bisect the angles. But $\backslash alpha\; -\; \backslash beta\; -\; (\backslash alpha\; -\; \backslash gamma)\; +\; \backslash beta\; -\; \backslash gamma\; =\; 0$; consequently (Art. 76) these lines meet in a point.

78. PROP. VII.—The perpendiculars from the angles of a triangle on the opposite sides meet in a point.

Let the angle between the lines whose equations are $\backslash alpha\; =\; 0,\; \backslash beta\; =\; 0$ be denoted by $(\backslash alpha\backslash beta)$, &c.; then (Art. 74) the equation to the perpendicular from the angle in which these two meet is

Similarly the equations to the other two perpendiculars are

Hence we have $u\; +\; v\; +\; w\; =\; 0$, therefore (Art. 76) the three lines meet in a point.

79. PROP. VIII.—If there be two triangles such that the perpendiculars from the angles of the one on the sides of the other meet in a point, then the perpendiculars from the angles of the latter on the sides of the former will also meet in a point.

Let the equations to the sides of the former triangle be $\backslash alpha\; =\; 0,\; \backslash beta\; =\; 0,\; \backslash gamma\; =\; 0$; and of the latter $\backslash alpha\text{'}\; =\; 0,\; \backslash beta\text{'}\; =\; 0,\; \backslash gamma\text{'}\; =\; 0$. Let also $(\backslash alpha\backslash beta)$ denote the angle between the lines $\backslash alpha\; =\; 0,\; \backslash beta\; =\; 0$; $(\backslash alpha\text{'}\backslash beta\text{'})$ between the lines $\backslash alpha\text{'}\; =\; 0,\; \backslash beta\text{'}\; =\; 0$, &c. Then (Art. 74) the equation to the perpendicular from the point $(\backslash alpha\backslash beta)$ on the line $\backslash gamma\text{'}\; =\; 0$ is $\backslash alpha\; \backslash cos(\backslash beta\backslash gamma\text{'})\; -\; \backslash beta\; \backslash cos(\backslash alpha\backslash gamma\text{'})\; =\; 0$....(1) and the equation to the perpendicular from the point $(\backslash alpha\backslash gamma)$ on the line $\backslash beta\text{'}\; =\; 0$ is $\backslash alpha\; \backslash cos(\backslash gamma\backslash beta\text{'})\; -\; \backslash gamma\; \backslash cos(\backslash alpha\backslash beta\text{'})\; =\; 0$....(2) and from $(\backslash beta\backslash gamma)$ on $\backslash alpha\text{'}\; =\; 0,\; \backslash beta\; \backslash cos(\backslash gamma\backslash alpha\text{'})\; -\; \backslash gamma\; \backslash cos(\backslash beta\backslash alpha\text{'})\; =\; 0$....(3) Now at the point where these three lines meet, $\backslash alpha$ is the same in (1) and (2), $\backslash beta$ in (1) and (3), and $\backslash gamma$ in (2) and (3). Hence by elimination between (1) and (3), we get

which, when combined with (2), gives

Now, if we change $\backslash alpha$ into $\backslash alpha\text{'}$, $\backslash beta$ into $\backslash beta\text{'}$, $\backslash gamma$ into $\backslash gamma\text{'}$, and conversely, we obtain the equations to the perpendiculars from the angles of the latter triangle on the sides of the former. At the point where the first and third of these lines intersect, we shall have, as before,

and the equation to the second line is

But because, by equation (4),

$\backslash cos(\backslash beta\text{'}\backslash gamma\text{'})\; \backslash cos(\backslash gamma\text{'}\backslash alpha\text{'})\; \backslash cos(\backslash alpha\text{'}\backslash beta)\; =\; \backslash cos(\backslash beta\text{'}\backslash alpha\text{'})\; \backslash cos(\backslash alpha\text{'}\backslash gamma\text{'})\; \backslash cos(\backslash gamma\text{'}\backslash beta)$, the same values of $\backslash alpha\text{'}$ and $\backslash gamma\text{'}$ satisfy both these equations. Hence the three lines meet in one point.

2. The Circle.

We shall confine ourselves to two properties only of the circle, both because of their more simple geometric deduction, and because they are included in the properties of the ellipse.

80. PROP. I.—If TPQ be a tangent to a circle; CT, BQ perpendiculars to the diameter from its extremities; then the rectangle PT.PQ = $\backslash text\{rad\}^2$.

When the centre is the origin, the equation to the circle is $x^2\; +\; y^2\; =\; a^2$ (Art. 10), and that of the tangent is $xx\text{'}\; +\; yy\text{'}\; =\; a^2$ (Art. 55).

81. PROP. II.—If through a fixed point chords are drawn to a circle, and tangents are drawn at the extremities of those chords, the locus of the points of intersection of the tangents is a straight line.

Let $b,\; c$ be the co-ordinates of the fixed point, $x\text{'},\; y\text{'}$ the co-ordinates of a point in which two tangents to the circle intersect one another; then (Art. 57) the equation to the chord which joins the points of contact is $xx\text{'}\; +\; yy\text{'}\; =\; a^2$. But, by hypothesis, $b,\; c$ are co-ordinates of a point in this line, $\backslash therefore\; bx\text{'}\; +\; cy\text{'}\; =\; a^2$. Now this is a simple equation between $x\text{'}$ and $y\text{'}$. Hence the truth of the proposition.

3. The Parabola.

82. PROP. I.—To find the latus rectum of the parabola.

The latus rectum is the ordinate to the axis drawn through the focus.

By Art. 12 the equation to the parabola is $y^2\; =\; 4ax$. Now, when $x\; =\; a,\; y$ is half the chord through the focus; $\backslash therefore$ that half-chord² = $4a^2$, or the latus rectum = $4a$.

83. PROP. II.—To find the subtangent of the parabola.

The subtangent is the portion of the diameter included between the ordinate and the tangent.

Let PT be the tangent meeting the axis in T; MT is the subtangent.

The equation to the tangent PT is (Art. 55) $yy\text{'}\; =\; 2a(x\; +\; x\text{'})$.

Now, at the point T, $y\; =\; 0,\; \backslash therefore\; x\; =\; -x\text{'}$, or AT = AM; hence the subtangent is double the abscissa.

84. COR.—Since AT = AM and AF = AD; by addition, FT = DM = PQ = FP; $\backslash therefore\; \backslash angle\; FPT\; =\; \backslash angle\; FTP\; =\; \backslash angle\; QPT$, or the tangent bisects the angle between two lines drawn from the point of contact, the one to the focus, the other perpendicular to the directrix.

This is the property of the tangent employed in the treatise on CONIC SECTIONS, to which the reader is referred for the geometrical properties to which it leads.

85. PROP. III.—To find the equation to the normal of the parabola.

The normal is the straight line which is perpendicular to the tangent at the point of contact.

Let PT be the tangent at the point P, PG perpendicular to PT is the normal.

The equation to PT is

and the equation to a line perpendicular to this is (Art. 69)

Analytical Geometry. Now the point P, whose co-ordinates are $x,\; y$, is a point in this line; therefore

and by subtraction $y\; -\; y\text{'}\; =\; -\backslash frac\{y\}\{2a\}(x\; -\; x\text{'})$, which is the equation to the normal.

86.—COR. At the point G, $y\; =\; 0$, $x\; =\; AG$. Hence $AG\; =\; x\; +\; 2a\; =\; AM\; +\; 2a$; $\backslash therefore\; MG\; =\; 2a$.

The line MG is called the subnormal; and it follows that the subnormal is equal to half the latus rectum.

87. PROP. IV.—The perpendicular from the focus on the tangent intersects the tangent in the axis of $y$.

The equation to the tangent is

and the equation to the perpendicular from F on the tangent is

At their point of section $x$ and $y$ are the same in both. Now if $x$ be not 0, $y\; -\; \backslash frac\{y\text{'}\}\{2\}$ is positive in equation (1) and negative in equation (2), or vice versa, which is impossible; $\backslash therefore\; x\; =\; 0$, or the point of intersection is in the axis of $y$.

88. COR.—$FP\; :\; FY\; ::\; FY\; :\; FA$,

89. PROP. V.—To find the locus of the intersection of the tangent with the perpendicular on it from the vertex.

The equation to the tangent is $yy\text{'}\; =\; 2a(x\; +\; x\text{'})$, $\backslash dots\; \backslash dots\; (1)$ and the equation to the perpendicular on it from the vertex is $y\; =\; -\backslash frac\{y\text{'}\}\{2a\}x$; $\backslash dots\; \backslash dots\; \backslash dots\; (2)$

Also the equation to the parabola is $y^2\; =\; 4ax\text{'}$, $\backslash dots\; \backslash dots\; (3)$

From equation (2), $y\; =\; -2a\backslash frac\{y\text{'}\}\{x\}$, and hence, from equation (1)

$2ax\text{'}\; =\; -\backslash frac\{2ay^2\}\{x\}\; -\; 2ax$. Substituting these values in (3), there is obtained

which shows (Art. 19) that the locus is a cissoid, the diameter of whose generating circle is AD.

90. PROP. VI.—To find the equation to the parabola when the axes are any diameter and the tangent at its extremity.

Let PN be the diameter parallel to AM; QN the ordinate parallel to the tangent at P; $x,\; y$ the co-ordinates of P; $x\text{'},\; y\text{'}$ of Q referred to the axis; $x\text{'}\text{'},\; y\text{'}\text{'}$ the co-ordinates PN, NQ referred to the diameter PN and the tangent PN.

Hence, $x\; =\; x\text{'}\; +\; x\text{'}\text{'}\; +\; y\text{'}\; \backslash cos\; T$, $y\; =\; y\text{'}\; +\; y\text{'}\text{'}\; \backslash sin\; T$;

therefore the equation $y^2\; =\; 4ax$ becomes

Now $\backslash tan\; T\; =\; \backslash frac\{2a\}\{y\text{'}\}$ (Art. 6); $\backslash therefore\; 2y\text{'}y\text{'}\text{'}\; \backslash sin\; T\; =\; 4ay\text{'}\text{'}\; \backslash cos\; T$; and $y\text{'}^2\; =\; 4ax\text{'}$; hence the equation is reduced to $y\text{'}\text{'}^2\; \backslash sin^2\; T\; =\; 4ax\text{'}\text{'}$.

But $\backslash frac\{4a\}\{\backslash sin^2\; T\}\; =\; 4FP$ (Art. 88); hence $y\text{'}\text{'}^2\; =\; 4FPx\text{'}\text{'}$ is the equation required.

The equation when the curve is referred to any diameter and its bisected chords is consequently of the same form as the equation when the curve is referred to the axis. The conclusions arrived at in the one case are very easily adapted to the other.

91. PROP. VII.—The semi-latus-rectum is an harmonic mean between the segments of any chord drawn through the focus.

Let PFR be a chord through the focus; then (Art. 13),

Now, $\backslash cos\; PFM\; +\; \backslash cos\; RFM\; =\; 0$; $\backslash therefore\; \backslash frac\{1\}\{FP\}\; +\; \backslash frac\{1\}\{FR\}\; =\; \backslash frac\{2\}\{2a\}$, which proves the proposition.

4. The Ellipse.

92. PROP. I.—If a circle be described about the axis major, then ordinates to the ellipse and the circle to the same abscissa, have to one another the proportion of the axis minor to the axis major.

Let MPQ be an ordinate to the abscissa CM; then

93. PROP. II.—To find the subtangent of the ellipse.

The equation to the tangent PT is $a^2yy\text{'}\; +\; b^2xx\text{'}\; =\; a^2b^2$. At the point T, $y\; =\; 0$; $\backslash therefore\; xx\text{'}\; =\; a^2$, or CT.CM = CA²; whence the subtangent

94. COR.—If a tangent be drawn to the circle at Q, it will cut the axis in the same point T, for CT is independent of b.

Analytical Geometry. area is $4\; a\; b\; \backslash sin\; (\backslash theta\; +\; \backslash phi)$
$=\; 4\; a\; b\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash phi\; +\; 4\; a\; b\; \backslash cos\; \backslash theta\; \backslash sin\; \backslash phi$
$=\; 4\; \backslash text\{\; PM\}\; \backslash cdot\; \backslash text\{CN\}\; +\; 4\; \backslash text\{\; CM\}\; \backslash cdot\; \backslash text\{DN\}$
$=\; 4\; \backslash frac\{a\}\{b\}\; \backslash text\{\; PM\}^2\; +\; 4\; \backslash frac\{b\}\{a\}\; \backslash text\{\; CM\}^2$ (Art. 103)
$=\; 4\; a\; b\; \backslash left(\; \backslash frac\{\backslash text\{PM\}^2\}\{b^2\}\; +\; \backslash frac\{\backslash text\{CM\}^2\}\{a^2\}\; \backslash right)\; =\; 4\; a\; b$.

107. PROP. IX.—The ellipse is the curve generated by a point whose distance from the focus is to its distance from the directrix as $e:1$.

Let $\backslash text\{EQ\}$ be the directrix at a distance from the centre $\backslash text\{CE\}\; =\; \backslash frac\{a\}\{e\}$; then

(Art. 15); $\backslash therefore\; \backslash text\{FP\}\; :\; \backslash text\{PQ\}\; ::\; e\; :\; 1$.

5. The Hyperbola.

The equation to the hyperbola differs from that to the ellipse only in having $-b^2$ in place of $b^2$. With this limitation, the properties given above and their demonstrations are common to both curves, and it would be superfluous to repeat them. We shall, accordingly, confine ourselves to that property of the hyperbola which has no parallel in the ellipse—the asymptote and its consequences.

108. PROP. I.—To find the equation to the asymptote of the hyperbola.

We have already (Art. 30) shown that if the values of $y$ be expressed in a descending series of powers of $x$, the equation to the asymptote may be obtained from it by omitting all negative powers.

Now in the hyperbola (Art. 17)—

Hence the straight line whose equation is $y\; =\; \backslash pm\; \backslash frac\{b\}\{a\}\; x$ is an asymptote to the curve.

If the curve be referred to conjugate diameters, the same reasoning will show that the equation to the asymptotes is

It is evident that the asymptotes are the diagonals of a parallelogram of which the conjugate diameters are the lines joining the points of bisection of opposite sides.

109. PROP. II.—If any chord of the hyperbola be terminated by the asymptotes, the rectangle by its segments is equal to the square of the semi-axis parallel to that chord.

Let $\backslash text\{RPQH\}$ (fig. 27) be a chord terminated by the asymptotes; then $\backslash text\{RP\}\; \backslash cdot\; \backslash text\{PH\}\; =\; b^2$.

Let the curve be referred to conjugate diameters, of which one is parallel to $\backslash text\{RH\}$; its equation is

and the equation to the asymptote

Now $\backslash text\{MP\}\; =\; \backslash text\{MQ\}$ being respectively $\backslash pm\; \backslash frac\{b\}\{a\}\; \backslash sqrt\{x^2\; -\; a^2\}$, and

$\backslash therefore$ the above equation gives $\backslash text\{RP\}\; \backslash cdot\; \backslash text\{PH\}\; =\; b^2$.

110. PROP. III.—To find the equation to the curve when referred to the asymptotes as axes.

Draw $\backslash text\{NP\}$ parallel to $\backslash text\{CR\}$, and let $\backslash text\{CN\}\; =\; x\text{'}$, $\backslash text\{NP\}\; =\; y\text{'}$.

Suppose $\backslash text\{CM\}$ in the direction of the transverse* axis $=\; x$, $\backslash text\{MP\}$ perpendicular to it $=\; y$, the angle $\backslash text\{RCM\}\; =\; \backslash theta$; then $x\; =\; x\text{'}\; \backslash cos\; \backslash theta\; +\; y\text{'}\; \backslash sin\; \backslash theta$, $y\; =\; y\text{'}\; \backslash sin\; \backslash theta\; -\; x\text{'}\; \backslash cos\; \backslash theta$; $\backslash therefore\; a^2\; (y\text{'}\; -\; x\text{'})^2\; \backslash sin^2\; \backslash theta\; -\; b^2\; (x\text{'}\; +\; y\text{'})^2\; \backslash cos^2\; \backslash theta\; =\; -a^2\; b^2$.

111. PROP. IV.—If any tangent be produced to meet the two asymptotes, the area of the triangle contained by the tangent and the two asymptotes is always the same.

Let $\backslash text\{UPT\}$ be the tangent; its equation (Art. 55);

and as the angle $\backslash text\{TCU\}$ is constant, the area $\backslash text\{TCU\}$ is always the same, whatever be the point $\backslash text\{P\}$.

We cannot close this portion of our subject without acknowledging our obligations to Mr Salmon, whose masterly treatise can hardly be too highly recommended.

ANALYTICAL GEOMETRY OF THREE DIMENSIONS; OR SOLID CO-ORDINATE GEOMETRY.

1. As in plane co-ordinate geometry, the position of a point in a plane is denoted by its distances from two given lines, so in solid geometry the position of a point in space

* The terms transverse and conjugate are more appropriate than major and minor in the case of the hyperbolas.

is denoted by its distances from three given planes. The direction in which the distance from any one plane is measured is parallel to the line of intersection of the other two. The planes of reference are called the co-ordinate planes, and their lines of intersection the co-ordinate axes; the point in which the three intersect one another being the origin.

SECTION I.—PROPERTIES OF THE PLANE AND STRAIGHT LINE.

We shall at present confine ourselves to rectangular axes.

2. PROP. I.—To find the distance between two points in terms of their co-ordinates.

Let $Ox,\; Oy,\; Oz$ be the axes of co-ordinates; $P,\; Q$ the two points; $x,\; y,\; z$ the co-ordinates of $P$; $x\text{'},\; y\text{'},\; z\text{'}$ the co-ordinates of $Q$. Through $P,\; Q$ draw planes parallel to the co-ordinate planes of $xy,\; xz,\; yz$; it is evident that they will form a parallelepipedon, of which $PQ$ is the diagonal. Let $QM$ be the edge of this parallelepipedon parallel to $x$, $MN$ parallel to $y$, and $NP$ to $z$; then it is evident that $QM\; =\; x\; -\; x\text{'}$, $MN\; =\; y\; -\; y\text{'}$, $NP\; =\; z\; -\; z\text{'}$; and $QM$ being perpendicular to the plane $PMN$, the angle $QMP$ is a right angle.

3. COR. 1.—If $\backslash alpha,\; \backslash beta,\; \backslash gamma$ be the angles which $PQ$ makes with the axes of $x,\; y,\; z$, we shall have

and $\backslash cos^2\; \backslash alpha\; +\; \backslash cos^2\; \backslash beta\; +\; \backslash cos^2\; \backslash gamma\; =\; 1$; consequently the angles which a straight line makes with the axes are not all three arbitrary, but where two are given, the third is determined by the above equation.

4. COR. 2.—If we take the three equations, $x\; -\; x\text{'}\; =\; PQ\; \backslash cos\; \backslash alpha$, $y\; -\; y\text{'}\; =\; PQ\; \backslash cos\; \backslash beta$, $z\; -\; z\text{'}\; =\; PQ\; \backslash cos\; \backslash gamma$, and multiply the first by $x\; -\; x\text{'}$, the second by $y\; -\; y\text{'}$, and the third by $z\; -\; z\text{'}$, and add the results, we shall get $(x\; -\; x\text{'})\; \backslash cos\; \backslash alpha\; +\; (y\; -\; y\text{'})\; \backslash cos\; \backslash beta\; +\; (z\; -\; z\text{'})\; \backslash cos\; \backslash gamma\; =\; PQ$.

5. PROP. II.—To find the angle between two straight lines in terms of the angles which the lines make with the axes.

Draw through the origin two lines $OP,\; OQ$ parallel to the given lines, and let $OP$ make with the axes the angles $\backslash alpha,\; \backslash beta,\; \backslash gamma$, and $OQ$ the angles $\backslash alpha\text{'},\; \backslash beta\text{'},\; \backslash gamma\text{'}$.

Take $OP\; =\; 1$, $OQ\; =\; 1$, and let the angle $POQ$ be $\backslash theta$; then $x\; =\; OP\; \backslash cos\; \backslash alpha\; =\; \backslash cos\; \backslash alpha$, $y\; =\; \backslash cos\; \backslash beta$, &c. Now $PQ^2\; =\; OP^2\; +\; OQ^2\; -\; 2\; OP\; \backslash cdot\; OQ\; \backslash cos\; \backslash theta$, $=\; 2\; -\; 2\; \backslash cos\; \backslash theta$.

6. PROP. III.—To find the equation to a plane.

Let $ABC$ be the plane, $P$ any point in it, of which the co-ordinates are $x,\; y,\; z$; $OQ$ perpendicular to the plane $=\; p$, making with the axes of $x,\; y$, and $z$ the angles $\backslash alpha,\; \backslash beta$, and $\backslash gamma$.

Let also $OP$ make with the axes the angles $\backslash alpha\text{'},\; \backslash beta\text{'},\; \backslash gamma\text{'}$; then, as in Art. 3, we have $x\; =\; OP\; \backslash cos\; \backslash alpha\text{'}$, $y\; =\; OP\; \backslash cos\; \backslash beta\text{'}$, $z\; =\; OP\; \backslash cos\; \backslash gamma\text{'}$.

7. COR. 1.—If $OA\; =\; a,\; OB\; =\; b,\; OC\; =\; c$, we have

hence $\backslash frac\{x\}\{a\}\; +\; \backslash frac\{y\}\{b\}\; +\; \backslash frac\{z\}\{c\}\; =\; 1$, is another form of the equation.

8. COR. 2.—Any simple equation in $x,\; y,\; z$ of which the general form is $Ax\; +\; By\; +\; Cz\; =\; D$ will represent a plane.

9. COR. 3.—If the equation be written in the form exhibited in the last corollary, we must have

10. COR. 4.—If $z\; =\; 0$ in Cor. 1, we get $\backslash frac\{x\}\{a\}\; +\; \backslash frac\{y\}\{b\}\; =\; 1$, which is obviously the equation to the line $AB$. This line is called the trace of the plane on the plane of $xy$. In like manner, $AC,\; BC$ are the traces of the plane on the planes of $xz$ and $yz$.

11. COR. 5.—If the plane be parallel to the axis of $z$ or perpendicular to the plane of $xy$, the line $OQ$ is in the plane of $xy$; $\backslash therefore\; \backslash gamma\; =\; \backslash frac\{\backslash pi\}\{2\}$ and $\backslash cos\; \backslash gamma\; =\; 0$; hence the equation to the plane becomes $x\; \backslash cos\; \backslash alpha\; +\; y\; \backslash cos\; \backslash beta\; =\; p$, or the equation to the plane is the same as the equation to its trace on the plane of $xy$.

12. PROP. IV.—To find the equations to a straight line.

A straight line is the intersection of two planes, consequently the equations to those planes will be satisfied simultaneously for the line. These are therefore the equations to the line.

If the two equations be $Ax\; +\; By\; +\; Cz\; =\; D$ and $A\text{'}x\; +\; B\text{'}y\; +\; C\text{'}z\; =\; D\text{'}$, we may obtain by elimination

as the form of the equations required.

13. The equations are sometimes obtained in the following manner:—

Let planes pass through the line respectively perpendicular to the planes of $xz$ and $yz$. Their equations will have

Analytical Geometry. the form (Art. 11) $x=az+p$, $y=bz+q$, which are the equations to the line.

14. COR. 1.—It is evident that if perpendiculars be drawn from every point in the given line to the plane of $xz$, they will intersect that plane in the line whose equation is $x=az+p$. This latter line is called the projection of the given line on the plane of $xz$. Consequently, the equations to a line are the equations to its projections on two of the co-ordinate planes.

15. PROP. V.—To express the angles which a straight line makes with the co-ordinate axes in terms of $a$ and $b$.

Let $PQ$, fig. of Art. 2, be the line, $x=az+p$, $y=bz+q$ its equations; then because the point $Q$, of which the co-ordinates are $x\text{'},\; y\text{'},\; z\text{'}$, is a point in the line, we have

which are the equations to a straight line in terms of the angles which it makes with the co-ordinate axes.

18. COR. 3.—If $x=a\text{'}z+p\text{'}$, $y=b\text{'}z+q\text{'}$ be the equations to another straight line; then, if $\backslash theta$ be the angle between them, we shall have

19. COR. 4.—In order that the two straight lines may be at right angles to one another, we must have $\backslash cos\; \backslash theta\; =\; 0$, or $aa\text{'}\; +\; bb\text{'}\; +\; 1\; =\; 0$.

20. PROP. VI.—To find the conditions that a straight line may be at right angles to a plane.

It is evident that the angles determined in Art. 9, which the perpendicular to the plane makes with the axes, must be equal to those determined in Art. 15, which the straight line makes with them; or

$\backslash therefore\; A\; =\; aC$, $B\; =\; bC$ are the conditions required.

21. PROP. VII.—To find the conditions that a straight line may coincide with a plane.

Let $x=az+p$, $y=bz+q$ be the equations to the line; $Ax+By+Cz=D$ the equation to the plane; then, since they coincide, we must have $A(az+p)\; +\; B(bz+q)\; +\; Cz\; =\; D$, whatever be $z$; hence $Aa+Bb+C=0$ and $Ap+Bq=D$ are the conditions required.

22. COR.—The condition that the straight line may be parallel to the plane is $Aa+Bb+C=0$; being the con-

dition that a line and plane parallel to these through the origin shall coincide.

23. PROP. VIII.—To find the length of the perpendicular from a given point on a given plane.

Let $P$ be the point of which the co-ordinates are $x\text{'},\; y\text{'},\; z\text{'}$; $PR$ perpendicular to the plane whose equation is $Ax+By+Cz=D$.

Draw $PQ$ parallel to the axis of $z$, meeting the plane in $Q$; then $PR\; =\; PQ\; \backslash cos\; QPR$

24. If from every point in the boundary of a given surface, straight lines be drawn parallel to a given line and meeting a fixed plane, they form the boundary of the projection of the surface on that plane. When the lines are drawn perpendicular to the plane, the projection is called an orthogonal projection. It is with this class alone that we are concerned here.

PROP. IX.—The projection of a plane area is to the area itself, as the cosine of the angle between the two planes is to unity.

Suppose the area divided into triangles, indefinitely small if necessary. Let $ABC$

be one of these triangles, $DEF$ its projection; and let the line of intersection of the planes of $ABC$, $DEF$ be $GK$. Through $A,\; B,\; C$ draw planes at right angles to $GK$, and therefore passing through $D,\; E$, and $F$; each of the angles $AGD,\; PHQ,\; CKF$ is equal to the angles between the planes.

Now the triangles $ABP,\; DEQ$ have $BP,\; EQ$ as their bases, and $GH$ as their common altitude;

Similarly of the other triangles. Hence the truth of the proposition.

25. PROP. X.—The square of any plane area is equal to the sum of the squares of its projections on the three co-ordinate planes.

Let $A\_x,\; A\_y,\; A\_z$ represent the projections of $A$ on the planes of $yz,\; xz$, and $xy$; $\backslash alpha,\; \backslash beta,\; \backslash gamma$, the angles which the perpendicular on $A$ makes with the axes of $x,\; y$, and $z$; then $\backslash alpha$ is the angle which the area makes with the plane of $yz$. $\backslash therefore\; A\_x\; =\; A\_y\; \backslash cos\; \backslash alpha$. Similarly $A\_y\; =\; A\; \backslash cos\; \backslash beta$, $A\_z\; =\; A\; \backslash cos\; \backslash gamma$. Now $\backslash cos^2\; \backslash alpha\; +\; \backslash cos^2\; \backslash beta\; +\; \backslash cos^2\; \backslash gamma\; =\; 1$; $\backslash therefore\; A\_x^2\; +\; A\_y^2\; +\; A\_z^2\; =\; A^2$.

26. PROP. XI.—To find the equation to a plane in terms of the area and its projections on the co-ordinate planes.

As in the last proposition $A\; =\; A\_x\; \backslash cos\; \backslash alpha$, &c.

Now (Art. 6) the equation to the plane is
$x\; \backslash cos\; \backslash alpha\; +\; y\; \backslash cos\; \backslash beta\; +\; z\; \backslash cos\; \backslash gamma\; =\; p$;
hence it becomes $x\; A\_x\; +\; y\; A\_y\; +\; z\; A\_z\; =\; p\; A$.

28. PROP. XII.—The projection of a straight line on a plane is to the straight line, as the cosine of the angle between them is to unity.

Let AB be the straight line meeting the plane in E; CD the projection; then AC, BD, which are perpendicular to ED, are in the plane BED at right angles to the plane of projection; and AEC or its equal BAF is the angle between the straight line and the plane; hence

29. PROP. XIII.—The projection of a straight line on another straight line is to the straight line itself, as the cosine of their angle of inclination is to unity.

When two lines are not in the same plane, their angle of inclination is the angle which one of them makes with a straight line drawn through it parallel to the other.

Let AB be the line to be projected; CD the line on which it is to be projected. Draw AE parallel to CD; then BAE is the angle of inclination of the lines. Through B draw the plane BFE perpendicular to CD, and therefore to AE; and draw AC parallel to EF. Then EFC, BFC, AEB are right angles, $\backslash therefore\; AF$ is a parallelogram and $AE\; =\; CF$ the projection of AB on CD. But
$AE\; :\; AB\; ::\; \backslash cos\; BAE\; :\; 1$;
$\backslash therefore\; CF\; :\; AB\; ::\; \backslash cos\; BAE\; :\; 1$.

30. PROP. XIV.—If any two points be connected both by a straight and by a broken line, the projection of the former on a given line is equal to the sum of the projections of every part of the latter.

The projection of PQ on the line AB is equal to the sum of the projections of PM, MN, NR, RQ. Through P, M, N, R, Q draw planes perpendicular to AB; the intercepted portions of AB are the projections of PM, MN, &c.; hence the proposition is evidently true provided none of the intercepted portions lap over each other or are negative, as would be the case with the projection of MN in the following figure. But the cosines employed will rectify this, provided we attend to the directions of the lines as indicated in the figure by the arrows. For the projection of MN will be negative, because the angle is greater than a right angle, whilst all the others will be positive, including that of RQ, for which the angle is negative, but its cosine positive.

31. By a similar process, and retaining the same restriction as to direction, we may show that if an area be connected with any number of areas, having common edges with the former area and with each other, the projection of the former on any plane is equal to the sum of the projections of all the others.

32. a.—The Sphere.

The characteristic property is that every point is equally distant from the centre.

Let $a,\; b,\; c$ be the co-ordinates of the centre, and $r$ the radius. The equation is evidently

33. b.—Oblique Cylinder on a Circular Base.

We shall give the simplest form of the equation, by supposing the circular base to be in the plane of $xy$ with its centre at the origin, so that its equation is $x^2\; +\; y^2\; =\; r^2$. This circle is called the directrix. A straight line moves parallel to itself in such a way as always to pass through the circumference of this circle. This line is called the generatrix. Let $x\; =\; az\; +\; p$, $y\; =\; bz\; +\; q$ be its equation. Then $a,\; b,\; r$ are known constants. Now $p,\; q$ are the values of $x$ and $y$ when $z=0$; they are therefore the co-ordinates of a point in the circumference of the directrix, or $p^2\; +\; q^2\; =\; r^2$, i. e. $(x-az)^2\; +\; (y-bz)^2\; =\; r^2$, which is the equation required.

34. CON.—A similar process will determine the equation to the cylinder whatever be the equation to the directrix.

35. c.—Oblique Cone on a Circular Base.

We shall suppose the circle to be the same as in Art. 33. The generatrix is now required to pass through a fixed point called the vertex of the cone. Let $a,\; b,\; c$ be the co-ordinates of this point; $x\; =\; Az\; +\; p$, $y\; =\; Bz\; +\; q$, the equations to the generatrix: then we must have $a\; =\; Ae\; +\; p$, $b\; =\; Be\; +\; q$, from which equations we obtain

and, since $p^2\; +\; q^2\; =\; r^2$, the equation required is $(az\; -\; ex)^2\; +\; (bz\; -\; ey)^2\; =\; r^2(z\; -\; c)^2$.

36. d.—The Ellipsoid.

The directrices are two ellipses at right angles to each other, having a common axis. We shall suppose their planes to be those of $xy$ and $xz$. The generatrix is a variable ellipse, such that its semi-axes are the ordinates of the directrices.

Let $\backslash frac\{x^2\}\{a^2\}\; +\; \backslash frac\{y^2\}\{b^2\}\; =\; 1$ (1) be the equation to AOB,

$\backslash frac\{x^2\}\{a^2\}\; +\; \backslash frac\{z^2\}\{c^2\}\; =\; 1$ (2) the equation to AOC,

the two directrices; so that $OA\; =\; a$, $OB\; =\; b$, $OC\; =\; c$. The equation to the generatrix RPQ will be

But by equations (1) and (2) $MQ^2\; =\; b^2\; \backslash left(1\; -\; \backslash frac\{x^2\}\{a^2\}\backslash right)$, $MR^2\; =$

$c^2\; \backslash left(1\; -\; \backslash frac\{x^2\}\{a^2\}\backslash right)$. Hence (3) becomes

37. CON.—If $b=c$, the ellipsoid becomes a spheroid, prolate or oblate according as $b$ is less or greater than $a$.

The directrix is a parabola, which we shall suppose to be in the plane of $xz$, the axis of $x$ being its axis, so that its equation is $z^2\; =\; 4ax$. The generatrix is a variable circle in the plane of $yz$, such that its radius is always the ordinate of the parabola. The equation to the generatrix is $y^2\; +\; z^2\; =\; \backslash text\{radius\}^2$, $\backslash therefore\; y^2\; +\; z^2\; =\; 4ax$ is the equation required.

This surface differs from the ellipsoid only in having the directrices two hyperbolas, with a common conjugate axis. Hence $a^2$ is negative, and the equation is therefore

This surface differs from the ellipsoid only in having the directrices hyperbolas with their transverse axis common. Hence $b^2$ and $c^2$ are negative, and the equation is therefore

The directrices are here two parabolas placed as the two ellipses were in the case of the ellipsoid. The generatrix is a variable ellipse as before.

In the figure of last Article, if we suppose AB, AC to be two parabolas whose equations are $MQ^2\; =\; 4ax$, $MR^2\; =\; 4bx$, the origin being at A, we shall have as the equation to the ellipse RPQ

The directrices are the same as in the last Article, but the generatrix is a variable hyperbola, whose semi-axes are the ordinates of the two parabolas. The equation to the hyperbola is therefore

Hence the equation to the surface is

Hitherto we have supposed the system of co-ordinates to be rectangular. This is indeed the fundamental system; but for the solution of certain problems it is necessary to pass to an oblique system. The investigation of the requisite formulae will constitute the subject of the present section.

Retaining the figure of Art. 2, and supposing the axes not to be at right angles, we have

Now $MP\; \backslash cos\; QMP$ is the projection of $MP$ on the line $MQ$, and is therefore (Art. 29) equal to the sum of the projections of $MN$ and $NP$ on the same line, i.e., to $MN\; \backslash cos\; NMQ\; +\; NP\; \backslash cos\; AMQ$. But if $\backslash alpha,\; \backslash beta,\; \backslash gamma$ be the angles $yOz,\; xOz,\; xOy$, it is evident that $MNP,\; AMQ$, and $NMQ$ are their respective supplements; hence

It is evident that this is effected by writing $x\text{'}\; +\; a,\; y\text{'}\; +\; b,\; z\text{'}\; +\; c$ for $x,\; y,\; z$.

Let $OM\; =\; x,\; MN\; =\; y,\; NP\; =\; z;\; OM\text{'}\; =\; x\text{'},\; M\text{'}N\text{'}\; =\; y\text{'},\; N\text{'}P\text{'}\; =\; z\text{'}$. If the line $OP$ be projected on the line $Ox$, the projection is $OM$ or $x$; and if the broken line $OM\text{'},\; M\text{'}N\text{'},\; N\text{'}P\text{'}$ be projected on $Ox$, the projection is equal to

care being taken, as in Art. 30, to take the angles $x\text{'}x$, &c. between the positive axis of $x\text{'}$ and the positive axis of $x$.

Of the nine angles $x\text{'}x,\; y\text{'}y,\; z\text{'}z$ employed above, six only are independent of one another; since by Art. 3 we have the three following relations between them, as conditions to be satisfied, viz.:

46. COR.—If the new axes are rectangular as well as the old, we have three other equations of condition amongst the angles, to be satisfied, viz.: $-\backslash cos\; x\text{'}y\; =\; 0,\; \backslash cos\; x\text{'}z\; =\; 0,\; \backslash cos\; y\text{'}z\; =\; 0$, which give (Art. 5)

These make up the six equations of condition.

To avoid the six equations of condition exhibited above, Euler devised the following beautiful method of transformation, which is of frequent use in the solution of Mechanical Problems.

Let the angle between the planes of $xy$ and $x\text{'}y\text{'}$ be called $\backslash theta$; let also these two planes intersect in the line $Ox\_1$, which makes with the axis of $x$ the angle $xOx\_1\; =\; \backslash phi$, and with the axis of $x\text{'}$ the angle $x\text{'}Ox\_1\; =\; \backslash psi$.

(1.) Transform the co-ordinates to $x\_1y\_1z\_1$ in the plane of $xy$, the axis of $z$ remaining unchanged; there results (PLANE GEOMETRY, Art. 40)

2.) Transform the co-ordinates to $y\_2z\_2$ in the plane $yz$, the axis of $x\_1$ remaining unchanged; there results

(3.) Transform the co-ordinates to $x\text{'}\_1y\text{'}\_1z\text{'}\_1$ in the plane $x\_1y\_1z\_1$, the axis of $z\text{'}$ remaining unchanged; there results

By substitution, we obtain

48. COR.—If $\backslash psi\; =\; 0$,

formulae which are of frequent use.

49. PROP. V.—To transform the co-ordinates from rectangular to polar.

Let $OP$ (figure of Art. 45) = $r$, $POz\; =\; \backslash theta$, $NOx\; =\; \backslash phi$; then $x\; =\; ON\; \backslash cos\; \backslash phi\; =\; r\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash phi$, $y\; =\; r\; \backslash sin\; \backslash theta\; \backslash sin\; \backslash phi$, and $z\; =\; r\; \backslash cos\; \backslash theta$ are the requisite formulae.

SECTION IV.—TANGENT PLANES AND NORMALS TO SURFACES.

50. PROP. 1.—To find the equation to the tangent plane to a surface.

If such a plane exists, it must have the property that if a section of it and of the surface be made by a plane parallel to any of the co-ordinate planes, the section of the plane will be a tangent line to the section of the surface.

Let $x,\; y,\; z$ be the co-ordinates of the point of contact, $x\text{'},\; y\text{'},\; z\text{'}$ those of any point in the tangent plane; and let the equation to that plane be

Put $y\text{'}\; =\; y$ in the plane and in the surface; then, since the resulting line is a tangent to the resulting curve, we have

Similarly $q\; =\; \backslash frac\{dz\}\{dy\}$.

51. COR.—If the equation to the surface be written under the form $u\; =\; 0$; and $\backslash frac\{du\}\{dx\},\; \backslash frac\{du\}\{dy\},\; \backslash frac\{du\}\{dz\}$ represent the partial differential co-efficients of $u$ with respect to $x,\; y$, and $z$ respectively; since the total differential co-efficient of $u$ with

respect to $x$, when $y$ is constant, must = 0, we shall have

and the equation to the tangent plane becomes

52. PROP. II.—To find the equation to the normal to a surface.

Retaining the notation of Art. 51; since the normal is perpendicular to the surface, and therefore to the tangent plane; if its equations are $x\text{'}\; -\; x\; =\; a(z\text{'}\; -\; z)$, $y\text{'}\; -\; y\; =\; b(z\text{'}\; -\; z)$, we must have (Art. 20)

whence the equations to the normal are,

53. COR.—The equations may also be written

54. EXAMPLE.—To find the equation to the tangent plane to the ellipsoid.

and the equation to the tangent plane becomes

55. If $p$ be the perpendicular from the centre on the tangent; we have (Art. 23)

56. PROP. III.—To find the locus of the intersection of the tangent plane of the ellipsoid with the perpendicular on it from the centre.

The equation to the tangent plane is

Consequently the equations to the perpendicular on it from the centre are (Art. 20)

also

Analytical Geometry. which, by equation (1), gives

57. If through the normal to a surface different planes be drawn, their intersections with the surface will be plane curves, which will generally have different degrees of curvature depending on the direction of the tangent line; and, further, if other sections than those through the normal are drawn, they will determine curves which have different degrees of curvature for the same direction of the tangent line. We proceed to determine the radius of curvature of any such section.

58. PROP. I.—The sum of the reciprocals of the radii of curvature of any two normal sections at right angles to each other is constant.

Let the normal be taken as the axis of $z$, and consequently the tangent plane as the plane of $xy$.

Let also the angle which a section makes with the plane of $xz$ be $\backslash theta$.

It is evident that $\backslash frac\{dz\}\{dx\}\; =\; 0,\; \backslash frac\{dz\}\{dy\}\; =\; 0$. Put $r$ for $\backslash frac\{d^2\; z\}\{dx^2\},\; s$ for

$\backslash frac\{d^2\; z\}\{dx\; dy\},\; t$ for $\backslash frac\{d^2\; z\}\{dy^2\}$; and let $OM\; =\; k,\; MN\; =\; K$; then if $R$ be the radius of curvature of this section,

$R\; =\; \backslash frac\{1\}\{2\}\; \backslash text\{\; limit\; of\; \}\; \backslash frac\{ON^2\}\{PN\}\; =\; \backslash frac\{1\}\{2\}\; \backslash text\{\; limit\}$

of $\backslash frac\{k^2\; +\; K^2\}\{\backslash frac\{r\; k^2\}\{2\}\; +\; s\; k\; K\; +\; t\; \backslash frac\{K^2\}\{2\}\; +\; \backslash dots\}$

59. PROP. II.—To determine the sections of greatest and least curvature.

Retaining the notation of the last article, we have

which will be a maximum or a minimum, when

or $\backslash tan\; 2\backslash theta\; =\; \backslash frac\{2s\}\{r-t\}$; an equation which gives two values of $\backslash theta$,

viz., $\backslash theta\; =\; \backslash alpha,\; \backslash theta\; =\; 90^\backslash circ\; +\; \backslash alpha$, or the maximum and minimum sections are at right angles to each other. These are called the principal sections.

60. PROP. III.—The curvature of any normal section (measured by the reciprocal of the radius of curvature) is equal to the sum of the products of the curvatures of the two principal sections by the cosines of the angles which they respectively make with it.

Let one of the principal sections be the plane of $xz$, or let the two values of $\backslash theta$ be $\backslash theta\; =\; 0,\; \backslash theta\; =\; 90^\backslash circ$. Let also $\backslash rho,\; \backslash rho\text{'}$ be the two principal radii of curvature;

Also, since $\backslash tan\; 2\backslash theta\; =\; 0$, we have $s\; =\; 0$,

61. PROP. IV.—If through a given tangent line at any point, both a normal and an oblique section be made, the radius of curvature of the oblique section is equal to the projection on its plane of the radius of curvature of the normal section.

Let the axis of $x$ be the given tangent, the plane of $xz$ the normal section, that of $xz\text{'}$ the oblique section.

Let $OM\; =\; k,\; MP\; =\; z,\; MN\; =\; k\text{'},\; NP\text{'}\; =\; z\text{'},\; \backslash angle\; zOz\text{'}\; =\; \backslash theta,\; R$ the radius of curvature in the normal section $xz,\; R\text{'}$ in the oblique section $xz\text{'}$.

therefore $R\text{'}\; =\; R\; \backslash cos^2\; \backslash theta$.

This is called Meunier's Theorem, and it establishes the fact, that if the diameter of a circle in the plane of $yz$ be that of the circle of curvature of the normal section, the chord of the same circle in the direction of the oblique section will be the diameter of the circle of curvature of that section; and consequently if the diameter of a sphere be the diameter of curvature of any normal section, the chord of the same sphere made by an oblique section through the same tangent line will be the diameter of the circle of curvature of that section; or the sections of the sphere by planes through the same tangent line will all be circles of curvature of the corresponding sections of the surface; and conversely.

62. PROP. V.—To find the conditions that a section of a surface may have a contact of the second order with a circle.

We shall suppose the touching circle to be determined by the section of a sphere, the cutting plane which is common to it and the given surface being supposed to pass through their ordinate $z$, so as to have for its equation $y\text{'}\; -\; y\; =\; m(x\text{'}\; -\; x)$; or if $x\; +\; h,\; y\; +\; k$ are the co-ordinates of a point in the section of either surface near their point of contact, the increments are connected by the equation $h\; =\; mh$.

Let $f(x,\; y,\; z)\; =\; 0$ be the equation to the given surface; $(X\; -\; \backslash alpha)^2\; +\; (Y\; -\; \backslash beta)^2\; +\; (Z\; -\; \backslash gamma)^2\; =\; R^2$ the equation to the sphere of which the section has a contact of the second order with

the given surface at the point whose co-ordinates are $x,\; y,\; z$. Let $z\_1,\; Z\_1$ be the ordinates of points in the common section of the two surfaces near their point of contact; then at the point in the curve of which the co-ordinates are $x+h,\; y+k$, we have

and at the corresponding point in the circle,

But the requirement for a contact of the second order is, that to the extent of $h^2$ these two expressions shall be coincident.

The conditions to be fulfilled are, therefore,

63. PROP. VI.—To find the radius of curvature and the co-ordinates of the centre of a normal section.

By differentiation, we obtain from the equation to the sphere of which the section of the normal plane is the circle of curvature

Now the conclusion arrived at in Art. 61 enables us to apply the conditions of the last Article; for if the section of the sphere by one plane is a section of curvature, every

section is so. Hence if we write $\backslash frac\{dz\}\{dx\}\; =\; p,\; \backslash frac\{dz\}\{dy\}\; =\; q,\; \backslash frac\{d^2z\}\{dx^2\}\; =\; r,$

$\backslash frac\{d^2z\}\{dx\; dy\}\; =\; s,\; \backslash frac\{d^2z\}\{dy^2\}\; =\; t$, we shall have, by equations (1) and (2)

of the last Article compared with (4), (5), and (6) of the present,

and consequently equation (3) of the last Article gives

Also by equations (2) and (3) we have

and hence by equation (1)

Equations (7), (8), and (9) determine the co-ordinates of the centre; and equation (10) gives the value of the radius of curvature of the section.

SECTION VI.—THE GENERATION OF SURFACES BY THE MOTION OF LINES.

We propose in the present section to determine the ordinary and differential equations to cylindrical, conical, and conoidal surfaces, and to surfaces of revolution.

Cylindrical Surfaces.

64. To determine the ordinary equation, we adopt the definition, that cylindrical surfaces are generated by the motion of a straight line which always moves parallel to itself.

Let $x\; =\; az\; +\; p,\; y\; =\; bz\; +\; q$ be the equations to the generating line, where $a$ and $b$ are the same for all the lines; and let $x\; =\; F(z),\; y\; =\; \backslash phi(z)$ be the equations to the directing curve: then at the points where the straight line meets the curve we have

$x\text{'}\; =\; az\text{'}\; +\; p,\; y\text{'}\; =\; bz\text{'}\; +\; q,\; x\text{'}\; =\; F(z\text{'}),\; y\text{'}\; =\; \backslash phi(z\text{'})$; from which equations we may eliminate $x,\; y$, and $z$, and obtain a relation between $p$ and $q$ and the constants $a$ and

$b$. Let this relation be $p\; =\; f(q)$; i.e.

the equation required.

65. To obtain the differential equation, we observe that a tangent plane always touches the curve along one of the generating lines. Hence the line whose equation is $x\; =\; az\; +\; p,\; y\; =\; bz\; +\; q$ is parallel to the plane whose equation is (Art. 50)

The condition is (Art. 22) $a\; \backslash frac\{dz\text{'}\}\{dx\}\; +\; b\; \backslash frac\{dz\text{'}\}\{dy\}\; =\; 1$, which is the differential equation required.

Conical Surfaces.

66. A conical surface is generated by a straight line which passes through a given point, and always meets a given curve.

Let $a,\; b,\; c$ be the co-ordinates of the given point; $x\; =\; F(z),\; y\; =\; \backslash phi(z)$, the equations to the given curve, the directrix; the equations to the generatrix are $x\; -\; a\; =\; m(z\; -\; c),\; y\; -\; b\; =\; n(z\; -\; c)$; also at the point where the line meets the directrix we shall have $x\text{'}\; -\; a\; =\; m(z\text{'}\; -\; c),\; y\text{'}\; -\; b\; =\; n(z\text{'}\; -\; c),\; y\text{'}\; =\; F(z\text{'}),\; y\text{'}\; =\; \backslash phi(z\text{'})$, from which equations $x\text{'},\; y\text{'},\; z\text{'}$ may be eliminated, and there will remain

the equation required.

67. To find the differential equation, it is only necessary to observe, that the tangent plane passes through the given

point; hence $z\; -\; c\; =\; \backslash frac\{dz\}\{dx\}(x\; -\; a)\; +\; \backslash frac\{dz\}\{dy\}(y\; -\; b)$ is the differential equation required.