gives \( \frac{b^2a}{3} = (\rho \times BD \times AD) \) for the content of the whole cone ABC: which appears from hence to be just \( \frac{1}{3} \) of a cylinder of the same base and altitude.

Prob. 4. To compute the surface of any solid body.

The fluxion of the surface of the solid is equal to the periphery of the surface, by whose motion the solid is generated, multiplied by its velocity on the edge of the solid, and the computation is made as in the foregoing.

Exam. Let it be proposed to determine the convex superficies of a cone ABC, fig. 11.

Then, the semi-diameter of the base (BD, or CD) being put \( = b \), the slanting line or hypotenuse AC \( = c \), and FH (parallel to DC) \( = y \), AG \( = z \), the surface \( = w \), its fluxion \( = w' \), and \( \rho \) = the periphery of a circle whose diameter is unity, we shall, from the similarity of the triangles ADC and Hmb, have \( b : c : : y : (w : x (Hb)) = \frac{c}{b} : \) whence \( w (z p y z) = \frac{2p^2}{b} \); and consequently \( w = \frac{2p^2}{b} \). This, when \( y = b \), becomes \( = pcb = p \times DC \times AC = \) the convex superficies of the whole cone ABC: which therefore is equal to a rectangle under half the circumference of the base and the slanting line.

The method of fluxions is also applied to find the centres of gravities, and oscillation of different bodies; to determine the paths described by projectiles and bodies acted on by central forces, with the laws of centripetal force in different curves, the retardates given to motions performed in resisting media, the attractions of bodies under different forms, the direction of wind which has the greatest effect on an engine, and to solve many other curious and useful problems.

FLY