{ "id": "e0b55fcc3e42cb6d4936cd48b44b3e5ca4d9a20f", "text": "XV. On the mathematical theory of suspension bridges, with tables for facilitating their construction. By Davies Gilbert, Esq. V.P.R.S. &c. Communicated March 9, 1826.\n\nRead March 9, 1826.\n\nMy attention was first directed to a consideration of suspension bridges, and of the catenary curve on which their theory depends, when the plan for making such a communication across the Menai Straits was submitted to the Commissioners appointed by Parliament to improve the communication by roads and bridges through Wales. It then appeared to me, that the proposed depth of curvature, was not sufficient for ensuring such a degree of strength and permanence as would be consistent with the due execution of a great national work. This opinion I advanced as a Member of the Commission. But wishing to take on myself the full responsibility for such increased expense, as must of necessity be occasioned by enlarging the curvature, I also printed some approximations, hastily deduced, in the Quarterly Journal of Science; and derived from them a confirmation of the opinion that had been given. The interval between the points of support and the road-way of the Menai Bridge has in consequence been augmented to fifty feet; and it now possesses that full measure of strength, which experience has established as requisite and sufficient for works of iron not perfectly at rest.\n\nSince bridges of suspension are obviously adapted to very general use, I have flattered myself with the hope of doing something serviceable to the public, by expanding into tables\nthe formulæ from which my approximations were derived; adding to them other formulæ and tables for the catenary of equal strength. A curve not merely of speculative curiosity, but of practical use, where a wide horizontal extent may chance to be combined with natural facilities for obtaining a correspondent height for the attachments.\n\nBoth the ordinary catenaries, and these of equal strength, like circles, parabolas, logarithmic curves, &c. have the property of being each identical with themselves in every respect but size: and as the radius, the parameter, and the subtangent give the respective magnitudes of these curves, so are the catenaries determined in magnitude by the tension (expressed in measures of the chain), which takes place at the middle point, or apex of the curve, where it is a minimum. Consequently, when this tension is determined or given, all the other relations may be expressed in the same manner as sines, cosines, &c. in the circle.\n\nI assume that the first principles of the catenary curve are known; they will, consequently, be noted with no other view, than to derive from them ulterior properties.\n\nFor the ordinary catenary:\n\nLet \\(a\\) = the tension at the apex, estimated in measures of the chain;\n\n\\(x\\) = the absciss, the versed sine, or depth of curvature;\n\n\\(y\\) = the ordinate, or semi-transverse length;\n\n\\(z\\) = the length of the curve.\n\nThen since the tension, \\((a,)\\) acts horizontally at the apex \\((A,)\\) since the weight of the chain \\((z)\\) acts at right angles to the former, and the force of suspension at \\((P)\\) acts in the direction of the tangent. These forces must be represented in direc-\ntion and in magnitude by the incremental triangle Prp; and\n\nAs \\( \\dot{x} : \\dot{y} :: z : a \\); as \\( \\dot{x}^2 : \\dot{y}^2 :: z^2 : a^2 \\);\n\nas \\( \\dot{x}^2 + \\dot{y}^2 : \\dot{x}^2 :: a^2 + z^2 : z^2 \\);\n\nBut \\( \\dot{x}^2 + \\dot{y}^2 = \\dot{z}^2 \\) universally.\n\nTherefore, \\( \\dot{x}^2 : \\dot{x}^2 :: a^2 + z^2 : z^2 \\); and \\( \\dot{x} = \\frac{z \\dot{z}}{\\sqrt{a^2 + z^2}} \\);\n\nconsequently, \\( x = \\sqrt{a^2 + z^2 - a} \\).\n\nEquation A\n\n\\[\n\\begin{align*}\n\\text{No. 1. } & x = \\sqrt{a^2 + z^2 - a} \\\\\n\\text{No. 2. } & z = \\sqrt{2ax + x^2} \\\\\n\\text{No. 3. } & a = \\frac{z^2 - x^2}{z \\cdot x}\n\\end{align*}\n\\]\n\nAgain; \\( \\dot{x} : \\dot{y} :: z : a \\); \\( \\dot{y} = \\frac{a \\dot{x}}{z} \\), substituting from Eq. A.\n\n\\[\n\\dot{y} = \\frac{a \\dot{x}}{\\sqrt{2ax + x^2}}; \\quad \\text{(No. 2. Equ. B. No. 1.)}\n\\]\n\n\\( y = a \\times \\text{natural log. of } \\frac{a + x + \\sqrt{2ax + x^2}}{a} = a \\times \\text{nat. log. } \\frac{a + x + z}{a} \\); or by substituting its value for \\( a \\) from Equ. A. No. 3, and dividing by \\( z + x \\).\n\nEqu. B. No. 2.) \\( y = a \\times \\text{nat. log. } \\frac{z + x}{z - x} \\);\n\nor, if \\( \\frac{z \\dot{z}}{\\sqrt{a^2 + z^2}} \\) be substituted for \\( \\dot{x} \\) in\n\n\\[\n\\dot{y} = \\frac{a \\dot{x}}{z}. \\quad \\dot{y} = a \\times \\frac{\\dot{z}}{\\sqrt{a^2 + z^2}}\n\\]\n\nand\n\nEqu. B. No. 3.) \\( y = a \\times \\text{nat. log. } \\frac{\\sqrt{a^2 + z^2 + z}}{a} \\).\n\nTo find \\( x \\) when \\( a \\) and \\( y \\) are given:\n\nLet \\( N = \\) the number of which \\( \\frac{y}{a} \\) (Equ. B. No. 1.) is the natural logarithm.\n\nThen \\( aN = a + x + \\sqrt{2ax + x^2} \\), and \\( \\sqrt{2ax + x^2} = aN - a - x \\),\n\nmake \\( aN - a = M \\). Then \\( 2ax + x^2 = M^2 - 2Mx + x^2 \\), and\n\nEqu. C.) \\( x = \\frac{M^2}{2M + 2a} \\),\n\n\\( x \\) being known.\n\n\\( z \\) is found from Equ. A. No. 2. and\nT, the tension at P, being obviously equal to $\\sqrt{a^2 + z^2}$,\nis equal (Eq. A. No. 2.) to $\\sqrt{a^2 + 2ax + x^2} = a + x$.\n\nThe angle of suspension is derived from the common ana-\nlogy of the incremental triangle, and of the forces corres-\nponding with it.\n\nTables I. and II. are constructed from these theorems, and\ntheir use will be best explained by an example.\n\nLet the span proposed for a suspension bridge be 800 feet,\nand let the adjunct weight of suspension rods, road-way, &c.\nbe taken at one-half of the weight of the chains; then, if the\nfull tenacity of iron is represented by the modulus of 14800\nfeet, the virtual modulus for the whole weight must be re-\nduced in the proportion of $2 + 1 : 2$, or to 9867 feet; and let\nit be determined to load the chains at the point of their\ngreatest strain, that is at the points of suspensions, with one-\nsixth part of the weight they are theoretically capable of\nsustaining.\n\nThen, since the semi-span is 400 feet, and $y$ in Table I. is\ntaken at an hundred measures, each of these measures must\nbe four feet, and the weight expressed in the same measures\nto be sustained at the points of suspension will be $9867 \\div 6 \\times 4$\n$= 411,125$. Now it appears from Table I. where $y$ is uni-\nformly an hundred, that when $T = 412$\n\n\\[\n\\begin{align*}\na &= 400 \\text{ measures or } 1600 \\text{ feet}, \\\\\nx &= 12.565 - - 50.260 \\\\\nz &= 101.045 - 404.180 \\\\\n&< \\text{ the angle of suspension } 75^\\circ 49'.\n\\end{align*}\n\\]\n\nHaving now determined $a$, the modulus, latus rectum, or\nparameter of the curve. In Table II. will be found all the\nrespective quantities for each measure of $y$. But as $a$ is in\n\nMDCCXXVI.\nthis table taken at an hundred measures, and it has been found to be 400 of the former, each measure here must be 4 times 4, or 16 feet; consequently, each gradation of \\( y \\) will also be 16 feet, and the whole semi-span \\( \\frac{400}{16} \\) or 25 measures. And since \\( z \\) will be given in the Table for each measure of \\( y \\), the adjunct weights may readily be adapted to a strict preservation of the catenary form.\n\nAt 21 measures of \\( y \\). \\( z = 21.1547 \\)\n\n20 measures of \\( y \\). \\( z = 20.1335 \\)\n\n\\[\n1.0212 \\times 16 = 16.3392 \\text{ feet.}\n\\]\n\nConsequently while the ordinate extends one measure, or 16 feet from the 20th to the 21st measure, the length of the curve will increase 16 feet and \\( \\frac{1}{3} \\) very nearly, and the adjunct weight should be increased in the same proportion.\n\nAt 21 the length of \\( x \\) is 2,2131 measures, or multiplied by 16 = 35,4096 feet, the length of the suspension rods to the level of the apex.\n\nIt appears from Table I. that the tension \\( T \\) for a given half span of 100 measures is very nearly at its minimum when \\( x = 65.85 \\) measures, almost one-third part of the whole span. In the example taken above \\( 65.85 \\times 4 = 263.4 \\) feet, an height not to be attained in practice, nor strictly applicable if it could be reached, because of the great length of suspension. If the span and height (\\( 2y \\) and \\( x \\)) were given, the other quantities would be found in a similar manner.\n\nIn the catenary of equal strength\n\n\\( a \\cdot x \\cdot y \\cdot z \\) remain as before; but another symbol must now be introduced, \\( \\zeta = \\) the mass of the chain. Then will the\nforces be represented as in the ordinary curve by the incremental triangle Prp. But now \\( \\dot{x} : \\dot{y} :: \\zeta : a \\). And by a repetition of the former steps \\( \\dot{x} = \\frac{\\zeta \\dot{z}}{\\sqrt{a^2 + \\zeta^2}} \\).\n\nBut on the principle of equal strength,\n\nAs \\( a : \\sqrt{a^2 + \\zeta^2} :: \\dot{z} : \\zeta \\)\n\ntherefore \\( \\dot{z} = a \\times \\frac{\\zeta}{\\sqrt{a^2 + \\zeta^2}} \\) and\n\nEqu. D.) \\( z = a \\times \\text{nat. log. } \\frac{\\sqrt{a^2 + \\zeta^2} + \\zeta}{a} \\);\n\nand by substituting \\( a \\times \\frac{\\zeta}{\\sqrt{a^2 + \\zeta^2}} \\) for \\( \\dot{z} \\) in the equation \\( \\dot{x} = \\frac{\\zeta \\dot{z}}{\\sqrt{a^2 + \\zeta^2}} \\)\n\n\\( \\dot{x} = a \\times \\frac{\\zeta}{a^2 + \\zeta^2} \\); consequently,\n\nEqu. E.) \\( x = \\frac{a}{2} \\times \\text{nat. log. } \\frac{a^2 + \\zeta^2}{a^2} \\).\n\nAgain, from the first analogy, \\( \\dot{y} = \\frac{a \\dot{z}}{\\zeta} \\),\n\nsubstitute for \\( \\dot{x} \\) its equal \\( a \\times \\frac{\\zeta}{a^2 + \\zeta^2} \\), and\n\n\\( \\dot{y} = a^2 \\times \\frac{\\zeta}{a^2 + \\zeta^2} \\); therefore,\n\nEqu. F.) \\( y = \\text{the cir. arc. of which } \\zeta \\text{ is the tangent to radius } a.\n\n\\( a \\) and \\( y \\) being given to find \\( \\zeta \\). Multiply \\( \\frac{y}{a} \\) by 57°,29578 (the tab. log. 1.7581226) and reduce the decimals of a degree into minutes and seconds; then will the tangent of that arc, multiplied by \\( a \\), be equal to \\( \\zeta \\).\n\nAnd when \\( \\zeta \\) has been determined, the other columns of Tables III. and IV. are constructed from the above theorems, in a manner perfectly similar to that used in calculating of Tables I. and II.; and they may be illustrated by the same example; observing that \\( a \\), now represents the uniform tension on each given magnitude of iron throughout the chains,\nand that the column T has the whole pull which any building or support may have to sustain in the direction of the tangent.\n\nIn Table III. \\( y \\) being, as before, an hundred measures of four feet each, \\( a \\) must be sought = 411,125, and by proportioning between 420 and 400\n\n\\[\n\\begin{align*}\nx &= 12.2904 \\\\\nz &= 101.0020 \\\\\n\\zeta &= 102.0235 \\\\\nT &= 423.6019\n\\end{align*}\n\\]\n\n\\[\n\\begin{align*}\n&- - - - 49.1616 \\\\\n&- - - - 404.0080 \\\\\n&- - - - 408.0940 \\\\\n&- - - - 1694.4076 \\\\\n< .76° 3' 17''.\n\\end{align*}\n\\]\n\n\\( a \\), or the modulus of this curve being fixed at 411,125 measures of 4 feet each, or at 1644.5 feet; and \\( a \\) in Table IV. being taken at 100 measures, each one will be 16,445 feet, and all the quantities are given for each gradation of \\( y \\).\n\nThus at 21 measures of \\( y \\). \\( z = 21.1564 \\) \\( \\zeta = 21.3142 \\)\n\n\\[\n\\begin{align*}\n20 \\text{ measures of } y. & \\quad z = 20.1347 \\quad \\zeta = 20.2710 \\\\\n1,0217 & \\quad 1,04332\n\\end{align*}\n\\]\n\n\\( 1.0217 \\times 16.445 = 16.8019 \\) feet the increase of \\( z \\);\n\n\\( 1.0432 \\times 16.445 = 17.1410 \\) feet the increase of material in \\( \\zeta \\):\n\nconsequently \\( \\frac{1.0432}{1.0217} = 1.021 \\), the quantity of matter in this part of the chain to maintain uniform strength, that at the apex being unity, and the adjunct matter should be in the proportion of 1 to 1,04332.\n\nMoreover \\( x \\) the versed sine, or the length of the suspension rods to the level of the apex will be at\n\n21 measures of \\( y \\). \\( x = 2.2214 \\) measures \\( \\times 16,445 = 36,531 \\) ft.\n\n20 measures of \\( y \\). \\( x = 2.0135 \\) measures \\( \\times 16,445 = 33,112 \\) ft.\nAssuming in the ordinary catenary that \\( x = 65.85 \\) measures, is the height of the attachment to give a maximum extent of span with any virtual tenacity of material, \\( a \\) will be 85 measures, and \\( a + x = 85 + 65.85 \\), or 150.85 measures equal the given virtual tenacity. This taken as before at \\( \\frac{2}{3} \\) of \\( \\frac{1}{6} \\) of 14800 feet, will give 10,875 feet for each measure, and the whole span at \\( 2y = 2175 \\) feet. Chains merely supporting themselves, and at the utmost of their tenacity will extend nine times further, or to 19575 feet.\n\nIn the catenary of equal strength, the semi-span being equal to the circular arc of which \\( \\zeta \\) is the tangent to radius \\( a \\), it is obvious that \\( a \\times \\text{semi-cir. arc} \\) must be the limit of the span. Therefore if \\( a = \\frac{2}{3} \\) of \\( \\frac{1}{6} \\) of 14800 feet, or 1644.44 \\( a \\times \\frac{c}{2} = 5154 \\) feet.\n\nAnd if the chains merely sustain themselves at their utmost tenacity, 5154 \\( \\times 9 \\) will give 46385 feet, equal to 8,785 miles, or somewhat more than 8 miles and three-quarters.\n\nBut this case is purely hypothetical, for the purpose of ascertaining a limit, since \\( \\zeta \\), the mass or weight of the chain must be infinite, and consequently its length: the figure approaching indefinitely near to that of a chain sustaining itself from an infinite height, which figure is identical with that of a building, capable so far as pressure and the strength of materials are alone concerned, of being carried to any elevation whatever. This figure is readily determined:\n\nLet \\( a = \\) the section of such a building at its base,\n\\( y = \\) the section at any height,\n\\( x = \\) that height;\nThen, since the section and the superincumbent pressure must always be in the same proportion to each other, \\( \\frac{x}{y} \\) and \\( \\frac{y}{x} \\) are in a constant ratio. Let then \\( \\frac{x}{m} = -\\frac{y}{m} \\) where \\( m \\) is the modulus of pressure in the given material; but when \\( x = 0, y = a \\), therefore \\( \\frac{x}{m} = \\text{the nat. log. } \\frac{a}{y} \\); or \\( \\frac{x}{A.m} = \\text{the tab. log. } \\frac{a}{y} \\). \\( A = 2,3025851 \\); but if \\( e \\) and \\( y \\) the homologous sides or diameters of these sections; then, \\( \\frac{x}{z.A.m} = \\text{tab. log. } \\frac{e}{y} \\).\n\nFinally, I would notice a correction of frequent use in practical surveying, to be deduced from the properties of the catenary curve.\n\nWhen the measuring chain is extended over ground uneven, intersected by ditches, or made soft by water, it cannot be laid flat, but must be elevated at both its extremities, while the middle just touches the surface: thus giving the measurement too great by the difference between the whole perifery and the double ordinate.\n\nLet \\( z = \\) the half length of the chain.\n\n\\( x = \\) the elevation at each end equal to the depths of curvature.\n\nThen Equ. B. No. 2. \\( y = a \\times \\text{nat. log. } \\frac{z+x}{z-x} \\),\n\nAnd Equ. A. No. 3. \\( a = \\frac{z^2-x^2}{2x} \\); therefore\n\n\\( y = \\frac{z^2-x^2}{2x} \\times \\text{nat. log. } \\frac{z+x}{z-x} \\).\n\nBut when \\( x \\) is very small in comparison of \\( z \\), the nat. log. of \\( \\frac{z+x}{z-x} \\) becomes \\( \\frac{2x}{z} \\), and\n\n\\( y = \\frac{z^2-x^2}{2x} \\times \\frac{2x}{z} = z - \\frac{x^2}{z} \\);\n\n\\( \\frac{x^2}{z} \\) is therefore the difference between half the chain and the\nordinate. If \\( x \\) be expressed in parts of the whole chain, \\( 4x^2 \\) will be the correction for the difference between the perifery and double ordinate.\n\nIf \\( x \\) (the elevation at each end) be one link of the common measuring chain, \\( 4x^2 = \\frac{1}{25} \\) of a link, \\( \\frac{1}{25} \\) of \\( \\frac{66}{100} \\) of a foot \\( = 0.3168 \\) of an inch, varying as the squares of \\( x \\).\n\nIf half the chain were considered as a straight line, and the hypotenuse of a right angled triangle, the horizontal distance would be \\( z - \\frac{x^2}{2z} \\), giving but one half of the true difference, \\( 0.1584 \\) parts of an inch.\n\nAnd if the chain were supposed to be in the arc of a circle, \\( z = y + \\frac{y^3}{6a^2} \\), &c. And \\( y = \\sqrt{2ax - x^2} \\) (when \\( x \\) is very small in comparison with \\( a \\)) \\( = \\sqrt{2ax} \\). Therefore \\( i = \\frac{y^2}{2x} \\).\n\nAnd since \\( y \\) is also small in comparison of \\( a \\), the second term of the series \\( \\left( \\frac{y^3}{6a^2} \\right) \\) will be the difference between the ordinate and the arc. Then substituting \\( \\frac{y^4}{4x^2} \\) for \\( a^2 \\),\n\n\\( \\frac{y^3}{6a^2} = \\frac{2x^2}{3y} \\); or if \\( x \\) be expressed in parts of the whole chain,\n\n\\( = \\frac{8}{3}x^2 \\) will be the whole correction, \\( = 0.2112 \\) parts of an inch, or two-thirds of the true difference.\n\nFormulae might readily be constructed for different elevations of the extremities of the chain, but they would prove much too complicated for practical use.\n\nOne further observation may be applicable to suspension bridges, wholly unconnected with the preceding investigations.\n\nIn the event of their wanting stability to counteract and restrain undulatory motion, the ballustrades may be carried\nto any required height, and rendered inflexible by diagonal braces; and if further means were required for imparting stability, such braces might be adjusted with screws to the suspension rods themselves, after these rods had acquired their exact positions, on the completion of the work.\n\n| \\( y = 100 \\) | Table I.— Ordinary Catenary. |\n|---|---|---|---|---|---|\n| \\( a \\) | N. | x. | z. | T. | Angle. |\n| 2000 | 1.051271 | 2.500511 | 100.041471 | 2002.500511 | 87° 8' 11\" |\n| 1950 | 1.052619 | 2.504593 | 100.042440 | 1952.504593 | 87° 3' 46\" |\n| 1900 | 1.054041 | 2.632163 | 100.045727 | 1902.632163 | 86° 59' 8\" |\n| 1850 | 1.055541 | 2.703298 | 100.047540 | 1852.703298 | 86° 54' 15\" |\n| 1800 | 1.057127 | 2.778421 | 100.050163 | 1802.778421 | 86° 49' 6\" |\n| 1750 | 1.058807 | 2.857914 | 100.054318 | 1752.857914 | 86° 43' 40\" |\n| 1700 | 1.060588 | 2.942018 | 100.057566 | 1702.942018 | 86° 37' 53\" |\n| 1650 | 1.062480 | 3.031204 | 100.060788 | 1653.031204 | 86° 31' 46\" |\n| 1600 | 1.064494 | 3.125974 | 100.064421 | 1603.125974 | 86° 25' 16\" |\n| 1550 | 1.066642 | 3.226852 | 100.068245 | 1553.226852 | 86° 18' 21\" |\n| 1500 | 1.068939 | 3.334558 | 100.073939 | 1503.334558 | 86° 10' 59\" |\n| 1450 | 1.071399 | 3.449618 | 100.078929 | 1453.449618 | 86° 3' 6\" |\n| 1400 | 1.074041 | 3.572907 | 100.084490 | 1403.572907 | 85° 54' 39\" |\n| 1350 | 1.076886 | 3.705344 | 100.090750 | 1353.705344 | 85° 45' 35\" |\n| 1300 | 1.079958 | 3.847958 | 100.097440 | 1303.847958 | 85° 35' 45\" |\n| 1250 | 1.083286 | 4.002035 | 100.105403 | 1254.002035 | 85° 25' 16\" |\n| 1200 | 1.086903 | 4.168981 | 100.114680 | 1204.168981 | 85° 13' 51\" |\n| 1150 | 1.090849 | 4.350543 | 100.125801 | 1154.350543 | 85° 1' 26\" |\n| 1100 | 1.095169 | 4.548545 | 100.137346 | 1104.548545 | 84° 47' 54\" |\n| 1050 | 1.099920 | 4.765440 | 100.150553 | 1054.765440 | 84° 33' 5\" |\n| 1000 | 1.105170 | 5.004084 | 100.165906 | 1005.004084 | 84° 16' 48\" |\n\nThe column in Table I. marked N (where the numbers equal \\( e^{\\frac{y}{a}} \\)) is given as the medium conducting to all the subsequent calculations.\nTable I. continued.—The Ordinary Catenary.\n\n| \\( y = 100 \\) | \\( a \\) | N. | \\( x \\) | \\( z \\) | T. | Angle. |\n|---|---|---|---|---|---|---|\n| 1000 | 1.105170 | 5.004084 | 100.165906 | 1005.004084 | 84° 16' 48\" |\n| 980 | 1.107428 | 5.106408 | 100.173025 | 985.106408 | 84° 9' 49\" |\n| 960 | 1.109785 | 5.213007 | 100.180582 | 965.213007 | 84° 2' 13\" |\n| 940 | 1.112247 | 5.324098 | 100.188974 | 945.324098 | 83° 54' 58\" |\n| 920 | 1.114822 | 5.440495 | 100.196191 | 925.440495 | 83° 47' 4\" |\n| 900 | 1.117519 | 5.561266 | 100.205825 | 905.561266 | 83° 38' 48\" |\n| 880 | 1.120344 | 5.687876 | 100.214837 | 885.687876 | 83° 30' 11\" |\n| 860 | 1.123309 | 5.820479 | 100.225255 | 865.820479 | 83° 21' 9\" |\n| 840 | 1.126423 | 5.959394 | 100.235949 | 845.959394 | 83° 11' 42\" |\n| 820 | 1.129698 | 6.105033 | 100.247321 | 826.105033 | 83° 1' 47\" |\n| 800 | 1.133148 | 6.258102 | 100.260296 | 806.258102 | 82° 51' 23\" |\n| 780 | 1.136785 | 6.418938 | 100.273356 | 786.418938 | 82° 40' 28\" |\n| 760 | 1.140627 | 6.588360 | 100.288153 | 766.588360 | 82° 28' 57\" |\n| 740 | 1.144691 | 6.767004 | 100.304328 | 746.767004 | 82° 16' 50\" |\n| 720 | 1.148996 | 6.955577 | 100.321527 | 726.955577 | 82° 4' 3\" |\n| 700 | 1.153504 | 7.154926 | 100.339869 | 707.154926 | 81° 50' 33\" |\n| 680 | 1.158422 | 7.366193 | 100.360075 | 687.366193 | 81° 36' 15\" |\n| 660 | 1.163595 | 7.590181 | 100.382517 | 667.590181 | 81° 21' 6\" |\n| 640 | 1.169118 | 7.828368 | 100.407143 | 647.828368 | 81° 5' 1\" |\n| 620 | 1.175025 | 8.081923 | 100.433570 | 628.081923 | 80° 47' 4\" |\n| 600 | 1.181360 | 8.352608 | 100.463404 | 608.352608 | 80° 29' 40\" |\n| 580 | 1.188169 | 8.642308 | 100.495985 | 588.642308 | 80° 10' 11\" |\n| 560 | 1.195508 | 8.952299 | 100.532176 | 568.952299 | 79° 49' 27\" |\n| 540 | 1.203419 | 9.283888 | 100.562366 | 549.283888 | 79° 27' 2\" |\n| 520 | 1.212043 | 9.645021 | 100.617335 | 529.645021 | 79° 2' 56\" |\n| 500 | 1.221402 | 10.033315 | 100.667683 | 510.033315 | 78° 36' 59\" |\n| 480 | 1.231625 | 10.454508 | 100.725490 | 490.454508 | 78° 8' 55\" |\n| 460 | 1.242830 | 10.912412 | 100.789382 | 470.912412 | 77° 38' 28\" |\n| 440 | 1.255172 | 11.412622 | 100.863052 | 451.412622 | 77° 5' 23\" |\n| 420 | 1.268829 | 11.961025 | 100.947150 | 431.961025 | 76° 29' 6\" |\n| 400 | 1.284025 | 12.565207 | 101.044792 | 412.565207 | 75° 49' 2\" |\n| 380 | 1.301032 | 13.233994 | 101.158163 | 393.233994 | 75° 5' 35\" |\n| 360 | 1.320192 | 13.978305 | 101.290757 | 373.978305 | 74° 17' 7\" |\n| 340 | 1.341941 | 14.812141 | 101.447796 | 354.812141 | 73° 32' 10\" |\n| 320 | 1.366837 | 15.752501 | 101.635337 | 335.752501 | 72° 22' 46\" |\n| 300 | 1.395612 | 16.821529 | 101.862069 | 316.821529 | 71° 14' 44\" |\n| 280 | 1.429239 | 18.047685 | 102.139232 | 298.047685 | 69° 57' 31\" |\n| 260 | 1.469049 | 19.458933 | 102.483745 | 279.458933 | 68° 29' 13\" |\n| 240 | 1.516896 | 21.126437 | 102.893226 | 261.126437 | 66° 47' 38\" |\n| 220 | 1.575420 | 23.118850 | 103.473548 | 243.118850 | 64° 48' 38\" |\n| 200 | 1.648721 | 25.525175 | 104.219022 | 225.525175 | 62° 28' 34\" |\n| 180 | 1.743908 | 28.559946 | 105.343499 | 208.559946 | 59° 39' 43\" |\n| 160 | 1.868245 | 32.280531 | 106.638654 | 192.280531 | 56° 19' 0\" |\n| 140 | 2.042722 | 37.258541 | 108.722538 | 177.258541 | 52° 10' 2\" |\n| 120 | 2.300975 | 44.134402 | 111.982596 | 164.134402 | 46° 58' 48\" |\n| 100 | 2.718281 | 54.308027 | 117.520071 | 154.308027 | 40° 23' 42\" |\n| 95 | 2.865180 | 57.674415 | 119.517684 | 152.674415 | 38° 28' 45\" |\n| 90 | 3.037731 | 61.511583 | 121.884206 | 151.511583 | 36° 26' 34\" |\n| 85 | 3.240907 | 65.521600 | 124.624934 | 150.521600 | 34° 17' 44\" |\n| 80 | 3.490342 | 71.073875 | 128.153485 | 151.073875 | 31° 58' 28\" |\n| 75 | 3.793667 | 77.147407 | 132.377616 | 152.147407 | 29° 32' 4\" |\n| 70 | 4.172733 | 84.433443 | 137.657866 | 154.433443 | 26° 57' 10\" |\n\nMDCCXXVI.\nTable II.—The Ordinary Catenary.\n\n| N. | y. | x. | z. | T. | Angle |\n|--------|-------|----------|----------|------------|-------|\n| 1.010050 | 1 | .004999 | 1.000000 | 100.004999 | 89°25'39\" |\n| 1.020201 | 2 | .020000 | 2.000100 | 100.020000 | 88°51'15\" |\n| 1.030454 | 3 | .045001 | 3.000398 | 100.045001 | 88°16'53\" |\n| 1.040810 | 4 | .080007 | 4.000992 | 100.080007 | 87°42'31\" |\n| 1.051271 | 5 | .125025 | 5.002074 | 100.125025 | 87°8'11\" |\n| 1.061836 | 6 | .180050 | 6.003540 | 100.180050 | 86°33'51\" |\n| 1.072508 | 7 | .245098 | 7.005701 | 100.245098 | 85°59'33\" |\n| 1.083287 | 8 | .320170 | 8.008520 | 100.320170 | 85°25'16\" |\n| 1.094174 | 9 | .405271 | 9.012128 | 100.405271 | 84°51'1\" |\n| 1.105170 | 10 | .500408 | 10.016591| 100.500408 | 84°16'48\" |\n| 1.116278 | 11 | .605609 | 11.022190| 100.605609 | 83°42'36\" |\n| 1.127496 | 12 | .720855 | 12.028744| 100.720855 | 83°8'37\" |\n| 1.138828 | 13 | .846186 | 13.036613| 100.846186 | 82°34'20\" |\n| 1.150273 | 14 | .981591 | 14.045708| 100.981591 | 82°0'14\" |\n| 1.161834 | 15 | 1.127107 | 15.056292| 101.127107 | 81°26'15\" |\n| 1.173510 | 16 | 1.282710 | 16.068289| 101.282710 | 80°52'17\" |\n| 1.185304 | 17 | 1.448471 | 17.081928| 101.448471 | 80°18'22\" |\n| 1.197217 | 18 | 1.624373 | 18.097326| 101.624373 | 79°44'31\" |\n| 1.209249 | 19 | 1.810427 | 19.114472| 101.810427 | 79°10'43\" |\n| 1.221402 | 20 | 2.006663 | 20.133536| 102.006663 | 78°36'59\" |\n| 1.233678 | 21 | 2.213114 | 21.154085| 102.213114 | 78°3'19\" |\n| 1.246076 | 22 | 2.429703 | 22.177836| 102.429703 | 77°29'43\" |\n| 1.258600 | 23 | 2.656680 | 23.203319| 102.656680 | 76°56'11\" |\n| 1.271249 | 24 | 2.893847 | 24.231042| 102.893847 | 76°22'45\" |\n| 1.284025 | 25 | 3.141302 | 25.261197| 103.141302 | 75°49'22\" |\n| 1.296929 | 26 | 3.399061 | 26.293838| 103.399061 | 75°16'5\" |\n| 1.309904 | 27 | 3.667187 | 27.329212| 103.667187 | 74°42'53\" |\n| 1.323129 | 28 | 3.945662 | 28.367237| 103.945662 | 74°9'46\" |\n| 1.336427 | 29 | 4.234542 | 29.408157| 104.234542 | 73°36'44\" |\n| 1.349838 | 30 | 4.533833 | 30.451966| 104.533833 | 73°3'48\" |\n| 1.363424 | 31 | 4.843577 | 31.498822| 104.843577 | 72°30'58\" |\n| 1.377127 | 32 | 5.163822 | 32.548877| 105.163822 | 71°58'13\" |\n| 1.390968 | 33 | 5.494589 | 33.602210| 105.494589 | 71°25'35\" |\n| 1.404947 | 34 | 5.835881 | 34.658818| 105.835881 | 70°53'3\" |\n| 1.419067 | 35 | 6.187768 | 35.718931| 106.187768 | 70°20'36\" |\n| 1.433329 | 36 | 6.550276 | 36.782633| 106.550276 | 69°48'18\" |\n| 1.447734 | 37 | 6.923431 | 37.849968| 106.923431 | 69°16'6\" |\n| 1.462284 | 38 | 7.307284 | 38.921115| 107.307284 | 68°44'0\" |\n| 1.476980 | 39 | 7.701863 | 39.996336| 107.701863 | 68°12'1\" |\n| 1.491824 | 40 | 8.107217 | 41.075182| 108.107217 | 67°40'10\" |\n| 1.506817 | 41 | 8.523379 | 42.158320| 108.523379 | 67°8'25\" |\n| 1.521961 | 42 | 8.950402 | 43.245697| 108.950402 | 66°36'48\" |\n| 1.537257 | 43 | 9.388315 | 44.337834| 109.388315 | 66°5'19\" |\n| 1.552706 | 44 | 9.837146 | 45.434453| 109.837146 | 65°33'57\" |\n| 1.568312 | 45 | 10.297011| 46.534188| 110.297011 | 65°2'43\" |\n| 1.584073 | 46 | 10.767851| 47.639448| 110.767851 | 64°31'46\" |\n| 1.599994 | 47 | 11.249817| 48.749582| 111.249817 | 64°0'39\" |\n| 1.616074 | 48 | 11.742877| 49.864522| 111.742877 | 63°29'49\" |\n| 1.632315 | 49 | 12.247092| 50.984407| 112.247092 | 63°59'7\" |\n| 1.648721 | 50 | 12.762587| 52.109512| 112.762587 | 62°28'34\" |\nTable II. continued.—The Ordinary Catenary.\n\n\\[a = 100.\\]\n\n| N. | y. | x. | z. | T. | Angle. |\n|------|--------|--------|--------|--------|--------|\n| 1.665290 | 51 | 13.289300 | 53.239600 | 113.289300 | 6° 58\" 9 |\n| 1.682027 | 52 | 13.827388 | 54.375311 | 113.827388 | 61 27 53 |\n| 1.698932 | 53 | 14.376853 | 55.516346 | 114.376853 | 60 57 45 |\n| 1.716006 | 54 | 14.937727 | 56.662872 | 114.937727 | 60 27 46 |\n| 1.733252 | 55 | 15.510107 | 57.815092 | 115.510107 | 59 57 56 |\n| 1.750672 | 56 | 16.094061 | 58.973138 | 116.094061 | 59 28 14 |\n| 1.768266 | 57 | 16.689588 | 60.137011 | 116.689588 | 58 58 42 |\n| 1.786037 | 58 | 17.290790 | 61.306900 | 117.290790 | 58 29 19 |\n| 1.803988 | 59 | 17.916770 | 62.483020 | 117.915770 | 58 0 5 |\n| 1.822118 | 60 | 18.546493 | 63.665306 | 118.546493 | 57 31 1 |\n| 1.840431 | 61 | 19.189099 | 64.854000 | 119.189099 | 57 2 5 |\n| 1.858927 | 62 | 19.835386 | 66.049113 | 119.835386 | 56 33 20 |\n| 1.877610 | 63 | 20.510098 | 67.250901 | 120.510098 | 56 4 43 |\n| 1.896480 | 64 | 21.186333 | 68.459366 | 121.188633 | 55 36 16 |\n| 1.915540 | 65 | 21.879300 | 69.674600 | 121.879300 | 55 7 59 |\n| 1.934792 | 66 | 22.582171 | 70.897028 | 122.582171 | 54 39 52 |\n| 1.954237 | 67 | 23.297283 | 72.126416 | 123.297283 | 54 11 54 |\n| 1.973877 | 68 | 24.024709 | 73.362990 | 124.024709 | 53 44 6 |\n| 1.993715 | 69 | 24.764560 | 74.606930 | 124.764560 | 53 16 28 |\n| 2.013752 | 70 | 25.510873 | 75.858326 | 125.510873 | 52 48 59 |\n| 2.033990 | 71 | 26.281725 | 77.117274 | 126.281725 | 52 21 41 |\n| 2.054433 | 72 | 27.059265 | 78.384034 | 127.059265 | 51 54 33 |\n| 2.075080 | 73 | 27.849426 | 79.658573 | 127.849426 | 51 27 34 |\n| 2.095935 | 74 | 28.652451 | 80.941048 | 128.652451 | 51 0 46 |\n| 2.117000 | 75 | 29.468327 | 82.231672 | 129.468327 | 50 34 8 |\n| 2.138276 | 76 | 30.297123 | 83.530476 | 130.297123 | 50 7 40 |\n| 2.159766 | 77 | 31.138056 | 84.837643 | 131.138056 | 49 41 22 |\n| 2.181472 | 78 | 31.993903 | 86.153296 | 131.993903 | 49 15 14 |\n| 2.203396 | 79 | 32.862044 | 87.477555 | 132.862044 | 48 49 16 |\n| 2.225540 | 80 | 33.743457 | 88.810542 | 133.743457 | 48 23 29 |\n| 2.247907 | 81 | 34.638263 | 90.154236 | 134.638263 | 47 57 52 |\n| 2.270500 | 82 | 35.546581 | 91.503418 | 135.546581 | 47 32 25 |\n| 2.293318 | 83 | 36.463871 | 92.863428 | 136.463871 | 47 7 8 |\n| 2.316366 | 84 | 37.403837 | 94.232762 | 137.403837 | 46 42 2 |\n| 2.339646 | 85 | 38.353056 | 95.611543 | 138.353056 | 46 17 6 |\n| 2.363160 | 86 | 39.310110 | 96.999880 | 139.310110 | 45 52 20 |\n| 2.386910 | 87 | 40.293084 | 98.397915 | 140.293084 | 45 27 45 |\n| 2.410900 | 88 | 41.284143 | 99.805856 | 141.284143 | 45 3 20 |\n| 2.435129 | 89 | 42.280243 | 101.236566 | 142.280243 | 44 39 5 |\n| 2.459602 | 90 | 43.308592 | 102.651607 | 143.308592 | 44 15 1 |\n| 2.484322 | 91 | 44.342313 | 104.080886 | 144.342313 | 43 51 7 |\n| 2.509290 | 92 | 45.390455 | 105.538544 | 145.390455 | 43 27 23 |\n| 2.533983 | 93 | 46.439931 | 106.967368 | 146.439931 | 43 4 18 |\n| 2.559981 | 94 | 47.539444 | 108.467655 | 147.539444 | 42 40 26 |\n| 2.585709 | 95 | 48.625206 | 109.948393 | 148.625206 | 42 17 13 |\n| 2.611696 | 96 | 49.720447 | 111.440152 | 149.720447 | 41 54 10 |\n| 2.637944 | 97 | 50.851184 | 112.943315 | 150.851184 | 41 31 18 |\n| 2.664455 | 98 | 51.988313 | 114.457186 | 151.988313 | 41 8 36 |\n| 2.691234 | 99 | 53.140537 | 115.982862 | 153.140537 | 40 46 4 |\n| 2.718281 | 100 | 54.308027 | 117.520072 | 154.308027 | 40 23 42 |\nTable III. The Catenary of equal strength.\n\n| \\( y = 100 \\) |\n|---|---|---|---|---|---|\n| \\( a \\) | \\( x \\) | \\( z \\) | \\( \\zeta \\) | \\( T \\) | Angle |\n| 1000 | 5.008288 | 100.166600 | 100.334300 | 1005.020800 | 0° 16' 13\" |\n| 980 | 5.110881 | 100.173640 | 100.348276 | 985.124220 | 84° 9' 12\" |\n| 960 | 5.217781 | 100.181250 | 100.363200 | 965.232000 | 84° 1' 54\" |\n| 940 | 5.329126 | 100.188850 | 100.378552 | 945.344276 | 83° 54' 16\" |\n| 920 | 5.445471 | 100.197071 | 100.395276 | 925.461072 | 83° 46' 19\" |\n| 900 | 5.566977 | 100.202654 | 100.413000 | 905.584230 | 83° 38' 1\" |\n| 880 | 5.694003 | 100.215333 | 100.432288 | 885.712432 | 83° 29' 20\" |\n| 860 | 5.827073 | 100.225792 | 100.452730 | 865.846882 | 83° 20' 15\" |\n| 840 | 5.966506 | 100.237329 | 100.475340 | 845.987772 | 83° 10' 44\" |\n| 820 | 6.112609 | 100.247806 | 100.497724 | 826.135404 | 83° 0' 45\" |\n| 800 | 6.266274 | 100.261054 | 100.523680 | 806.290880 | 82° 50' 16\" |\n| 780 | 6.427811 | 100.274590 | 100.551048 | 786.454344 | 82° 39' 15\" |\n| 760 | 6.598152 | 100.289657 | 100.580680 | 766.626896 | 82° 27' 40\" |\n| 740 | 6.777369 | 100.305695 | 100.613064 | 746.808518 | 82° 15' 25\" |\n| 720 | 6.966790 | 100.322732 | 100.647648 | 727.000673 | 82° 2' 32\" |\n| 700 | 7.167238 | 100.342923 | 100.685480 | 707.204050 | 81° 48' 53\" |\n| 680 | 7.379542 | 100.362168 | 100.720972 | 687.419752 | 81° 34' 26\" |\n| 660 | 7.604848 | 100.384645 | 100.772166 | 667.647826 | 81° 19' 7\" |\n| 640 | 7.844443 | 100.409125 | 100.821568 | 647.892736 | 81° 2' 51\" |\n| 620 | 8.099715 | 100.430355 | 100.876232 | 628.152876 | 80° 45' 31\" |\n| 600 | 8.370382 | 100.465909 | 100.930680 | 608.430840 | 80° 27' 2\" |\n| 580 | 8.663690 | 100.498855 | 101.002534 | 588.728710 | 80° 7' 17\" |\n| 560 | 8.976381 | 100.535447 | 101.079360 | 569.048704 | 79° 46' 7\" |\n| 540 | 9.312582 | 100.576282 | 101.158740 | 549.393354 | 79° 23' 23\" |\n| 520 | 9.675126 | 100.621836 | 101.250968 | 529.765704 | 78° 58' 53\" |\n| 500 | 10.067350 | 100.679481 | 101.362400 | 510.169400 | 78° 32' 27\" |\n| 480 | 10.552010 | 100.780247 | 101.472192 | 490.668864 | 78° 3' 48\" |\n| 460 | 10.956213 | 100.796941 | 101.605490 | 471.087748 | 77° 32' 39\" |\n| 440 | 11.402781 | 100.872044 | 101.757920 | 451.613044 | 76° 58' 41\" |\n| 420 | 12.018908 | 100.958305 | 101.933328 | 432.192558 | 76° 21' 29\" |\n| 400 | 12.630692 | 101.056700 | 102.136560 | 412.832200 | 75° 40' 33\" |\n| 380 | 13.312576 | 101.174410 | 102.373976 | 393.548520 | 74° 55' 19\" |\n| 360 | 14.071210 | 101.311236 | 102.653784 | 374.349582 | 74° 5' 4\" |\n| 340 | 14.922900 | 101.473699 | 102.986884 | 355.252222 | 73° 8' 53\" |\n| 320 | 15.886128 | 101.668413 | 103.387488 | 336.287040 | 72° 5' 42\" |\n| 300 | 16.949763 | 101.904940 | 103.875990 | 317.474760 | 70° 54' 5\" |\n| 280 | 18.250135 | 102.160162 | 104.480264 | 298.858028 | 69° 32' 14\" |\n| 260 | 19.729226 | 102.564124 | 105.241136 | 280.497074 | 67° 57' 47\" |\n| 240 | 21.405587 | 103.025715 | 106.219200 | 262.454784 | 66° 7' 36\" |\n| 220 | 23.555838 | 103.632647 | 107.507994 | 244.863168 | 63° 57' 23\" |\n| 200 | 26.116574 | 104.447443 | 109.260480 | 227.898480 | 61° 21' 7\" |\n| 180 | 29.336487 | 105.580330 | 111.739482 | 211.862484 | 58° 10' 8\" |\n| 160 | 33.525185 | 107.228464 | 115.437376 | 197.296208 | 54° 11' 24\" |\n| 140 | 39.241137 | 109.779803 | 121.380952 | 185.292618 | 49° 4' 28\" |\n| 120 | 47.626016 | 114.104417 | 132.093348 | 178.401912 | 42° 15' 12\" |\n| 100 | 61.562643 | 122.619114 | 155.740770 | 185.081570 | 32° 42' 15\" |\n| 95 | 66.748734 | 126.148321 | 166.629316 | 191.808059 | 29° 41' 19\" |\n| 90 | 73.141390 | 130.727676 | 181.797084 | 202.855668 | 26° 20' 16\" |\n| 85 | 81.314301 | 136.905055 | 204.267512 | 221.246059 | 22° 35' 35\" |\n| 80 | 92.332784 | 145.717467 | 240.765568 | 253.708616 | 18° 22' 48\" |\n| 75 | 108.536763 | 159.466590 | 309.878850 | 318.825817 | 13° 36' 20\" |\n| 70 | 136.703450 | 184.926359 | 488.855143 | 493.841432 | 8° 8' 56\" |\nTable IV.—The Catenary of equal strength.\n\n\\[a = 100.\\]\n\n| \\(y\\) | \\(x\\) | \\(z\\) | \\(\\xi\\) | \\(T\\) | Angle |\n|------|------|------|--------|------|-------|\n| 1 | .004999 | .999990 | 1.00001 | 100.00500 | 89° 25' 37\" |\n| 2 | .020003 | 2.000088 | 2.00022 | 100.020006 | 88° 51' 14\" |\n| 3 | .045005 | 3.000431 | 3.00088 | 100.045016 | 88° 16' 52\" |\n| 4 | .080021 | 4.001021 | 4.00208 | 100.080054 | 87° 42' 29\" |\n| 5 | .125046 | 5.002067 | 5.00415 | 100.125125 | 87° 8' 6\" |\n| 6 | .180107 | 6.003541 | 6.00714 | 100.180270 | 86° 33' 44\" |\n| 7 | .245198 | 7.005697 | 7.01143 | 100.245499 | 85° 59' 21\" |\n| 8 | .323389 | 8.008498 | 8.01706 | 100.320852 | 85° 24' 58\" |\n| 9 | .405548 | 9.012161 | 9.02436 | 100.406373 | 84° 50' 46\" |\n| 10 | .500828 | 10.016660 | 10.03343 | 100.502080 | 84° 16' 13\" |\n| 11 | .606218 | 11.022229 | 11.04456 | 100.608062 | 83° 41' 50\" |\n| 12 | .721234 | 12.028425 | 12.05789 | 100.723845 | 83° 7' 28\" |\n| 13 | .847386 | 13.030754 | 13.07372 | 100.850992 | 82° 33' 5\" |\n| 14 | .983205 | 14.045921 | 14.09215 | 100.980635 | 81° 58' 42\" |\n| 15 | 1.129248 | 15.050560 | 15.11351 | 101.135644 | 81° 24' 20\" |\n| 16 | 1.285490 | 16.068670 | 16.13791 | 101.293792 | 80° 49' 57\" |\n| 17 | 1.452011 | 17.082468 | 17.16567 | 101.462608 | 80° 15' 34\" |\n| 18 | 1.628815 | 18.097959 | 18.19691 | 101.642158 | 79° 41' 12\" |\n| 19 | 1.815961 | 19.115360 | 19.23197 | 101.832558 | 79° 6' 49\" |\n| 20 | 2.013470 | 20.134658 | 20.27097 | 102.033830 | 78° 32' 23\" |\n| 21 | 2.221395 | 21.150371 | 21.31424 | 102.246255 | 77° 58' 4\" |\n| 22 | 2.439770 | 22.170619 | 22.36191 | 102.469780 | 77° 23' 41\" |\n| 23 | 2.668051 | 23.205504 | 23.41433 | 102.704585 | 76° 49' 19\" |\n| 24 | 2.908061 | 24.233742 | 24.47164 | 102.950768 | 76° 14' 56\" |\n| 25 | 3.158106 | 25.264601 | 25.53424 | 103.208504 | 75° 40' 33\" |\n| 26 | 3.418774 | 26.297360 | 26.60212 | 103.477887 | 75° 6' 11\" |\n| 27 | 3.690164 | 27.334154 | 27.67581 | 103.759100 | 74° 31' 48\" |\n| 28 | 3.972311 | 28.373174 | 28.75540 | 104.052264 | 73° 57' 25\" |\n| 29 | 4.265294 | 29.415243 | 29.84128 | 104.357567 | 73° 23' 3\" |\n| 30 | 4.569158 | 30.460378 | 30.93360 | 104.675156 | 72° 48' 40\" |\n| 31 | 4.883083 | 31.508739 | 32.03269 | 105.005213 | 72° 14' 17\" |\n| 32 | 5.209839 | 32.560521 | 33.13891 | 105.347935 | 71° 39' 55\" |\n| 33 | 5.546782 | 33.615738 | 34.25243 | 105.703501 | 71° 5' 32\" |\n| 34 | 5.894915 | 34.674039 | 35.37366 | 106.072131 | 70° 31' 9\" |\n| 35 | 6.254281 | 35.737235 | 36.50280 | 106.454005 | 69° 56' 47\" |\n| 36 | 6.624997 | 36.803792 | 37.64030 | 106.849383 | 69° 22' 24\" |\n| 37 | 7.007106 | 37.874291 | 38.78626 | 107.258446 | 68° 48' 2\" |\n| 38 | 7.400749 | 38.948988 | 39.94126 | 107.681495 | 68° 13' 39\" |\n| 39 | 7.805067 | 40.027947 | 41.10545 | 108.118722 | 67° 39' 16\" |\n| 40 | 8.222888 | 41.111407 | 42.27931 | 108.570433 | 67° 4' 54\" |\n| 41 | 8.651889 | 42.199404 | 43.46038 | 109.036870 | 66° 30' 31\" |\n| 42 | 9.092196 | 43.292198 | 44.67242 | 109.518354 | 65° 56' 8\" |\n| 43 | 9.544771 | 44.389841 | 45.89509 | 110.015128 | 65° 21' 46\" |\n| 44 | 10.009478 | 45.492556 | 47.07804 | 110.527566 | 64° 47' 23\" |\n| 45 | 10.486371 | 46.600436 | 48.30547 | 111.042096 | 64° 13' 0\" |\n| 46 | 10.975622 | 47.713735 | 49.54487 | 111.606062 | 63° 38' 38\" |\n| 47 | 11.477312 | 48.832499 | 50.79655 | 112.161892 | 63° 4' 15\" |\n| 48 | 11.991595 | 49.957023 | 52.06108 | 112.740211 | 62° 29' 52\" |\n| 49 | 12.518572 | 51.08569 | 53.34078 | 113.335897 | 61° 55' 32\" |\n| 50 | 13.058418 | 52.223810 | 54.63024 | 113.949396 | 61° 21' 7\" |\nTable IV. continued.—The Catenary of equal strength.\n\n\\[ a = 100. \\]\n\n| y | x | z | ξ | T | Angle |\n|-----|-------|-------|------|------|-------|\n| 51 | 13.611226 | 53.366417 | 55.93584 | 114.581052 | 60° 46' 44\" |\n| 52 | 14.177189 | 54.515494 | 57.25618 | 115.231377 | 60° 12' 22\" |\n| 53 | 14.756401 | 55.676905 | 58.59167 | 115.900748 | 59° 37' 59\" |\n| 54 | 15.349977 | 56.833577 | 59.94296 | 116.589191 | 59° 3' 36\" |\n| 55 | 15.955315 | 58.002974 | 61.31049 | 117.298661 | 58° 29' 14\" |\n| 56 | 16.575346 | 59.170619 | 62.69495 | 118.028208 | 57° 54' 51\" |\n| 57 | 17.209276 | 60.363609 | 64.09582 | 118.778802 | 57° 20' 29\" |\n| 58 | 17.857313 | 61.555215 | 65.51678 | 119.510332 | 56° 46' 6\" |\n| 59 | 18.510976 | 62.754711 | 66.95554 | 120.345521 | 56° 11' 43\" |\n| 60 | 19.16491 | 63.962210 | 68.41302 | 121.162801 | 55° 37' 21\" |\n| 61 | 19.888020 | 65.178046 | 69.89186 | 122.003580 | 55° 2' 58\" |\n| 62 | 20.594400 | 66.402358 | 71.39084 | 122.868440 | 54° 28' 35\" |\n| 63 | 21.315910 | 67.635500 | 72.91145 | 123.758155 | 53° 54' 13\" |\n| 64 | 22.052701 | 68.877606 | 74.45432 | 124.673361 | 53° 19' 50\" |\n| 65 | 22.805074 | 70.129059 | 76.02042 | 125.614906 | 52° 45' 27\" |\n| 66 | 23.573186 | 71.389994 | 77.61043 | 126.583487 | 52° 11' 5\" |\n| 67 | 24.357371 | 72.660825 | 79.22540 | 127.580036 | 51° 36' 42\" |\n| 68 | 25.157787 | 73.941997 | 80.86608 | 128.605306 | 51° 2' 19\" |\n| 69 | 25.974778 | 75.233031 | 82.53360 | 129.660301 | 50° 27' 57\" |\n| 70 | 26.808551 | 76.535188 | 84.22878 | 130.745895 | 49° 53' 34\" |\n| 71 | 27.659459 | 77.848058 | 85.95285 | 131.863168 | 49° 19' 11\" |\n| 72 | 28.527710 | 79.172384 | 87.70674 | 133.013056 | 48° 44' 49\" |\n| 73 | 29.413697 | 80.508436 | 89.49175 | 134.196771 | 48° 10' 26\" |\n| 74 | 30.317647 | 81.856432 | 91.30890 | 135.415343 | 47° 36' 4\" |\n| 75 | 31.239989 | 83.216866 | 93.15964 | 136.670112 | 47° 1' 41\" |\n| 76 | 32.180960 | 84.589966 | 95.04510 | 137.962209 | 46° 27' 18\" |\n| 77 | 33.140961 | 85.975963 | 96.96618 | 139.293095 | 45° 52' 56\" |\n| 78 | 34.120421 | 87.375961 | 98.92611 | 140.664948 | 45° 18' 33\" |\n| 79 | 35.119618 | 88.789594 | 100.92453 | 142.076604 | 44° 44' 10\" |\n| 80 | 36.139051 | 90.214039 | 102.96381 | 143.532386 | 44° 9' 48\" |\n| 81 | 37.179043 | 91.660396 | 105.04542 | 145.032900 | 43° 35' 25\" |\n| 82 | 38.240111 | 93.118455 | 107.17133 | 146.579992 | 43° 1' 2\" |\n| 83 | 39.322622 | 94.592159 | 109.34320 | 148.175337 | 42° 26' 40\" |\n| 84 | 40.427139 | 96.082135 | 111.56319 | 149.821051 | 41° 52' 17\" |\n| 85 | 41.554052 | 97.588753 | 113.82816 | 151.518952 | 41° 17' 54\" |\n| 86 | 42.703987 | 99.112099 | 116.15555 | 153.271369 | 40° 43' 32\" |\n| 87 | 43.877350 | 100.654374 | 118.53239 | 155.083097 | 40° 9' 9\" |\n| 88 | 45.074822 | 102.214506 | 120.96637 | 156.948608 | 39° 34' 46\" |\n| 89 | 46.296874 | 103.793554 | 123.45986 | 158.878469 | 39° 0' 24\" |\n| 90 | 47.544231 | 105.392291 | 126.01578 | 160.872559 | 38° 26' 1\" |\n| 91 | 48.817411 | 107.011233 | 128.63685 | 162.933851 | 37° 51' 39\" |\n| 92 | 50.117199 | 108.651210 | 131.32634 | 165.065469 | 37° 17' 16\" |\n| 93 | 51.444173 | 110.312786 | 134.08729 | 167.270444 | 36° 42' 53\" |\n| 94 | 52.799201 | 111.996881 | 136.92343 | 169.552431 | 36° 8' 31\" |\n| 95 | 54.182891 | 113.704104 | 139.83816 | 171.914846 | 35° 34' 8\" |\n| 96 | 55.596244 | 115.435462 | 142.83573 | 174.361831 | 34° 59' 45\" |\n| 97 | 57.039914 | 117.191641 | 145.92002 | 176.897299 | 34° 25' 23\" |\n| 98 | 58.514946 | 118.973717 | 149.09580 | 179.525931 | 33° 51' 0\" |\n| 99 | 60.032087 | 120.782488 | 152.36759 | 182.252247 | 33° 16' 37\" |\n| 100 | 61.562647 | 122.619117 | 155.74077 | 185.081573 | 32° 42' 15\" |", "source": "olmocr", "added": "2026-01-12", "created": "2026-01-12", "metadata": { "Source-File": "/home/jic823/projects/def-jic823/royalsociety/pdfs/107810.pdf", "olmocr-version": "0.3.4", "pdf-total-pages": 18, "total-input-tokens": 27074, "total-output-tokens": 26680, "total-fallback-pages": 0 }, "attributes": { "pdf_page_numbers": [ [ 0, 0, 1 ], [ 0, 1550, 2 ], [ 1550, 3203, 3 ], [ 3203, 4978, 4 ], [ 4978, 6647, 5 ], [ 6647, 8282, 6 ], [ 8282, 10035, 7 ], [ 10035, 11665, 8 ], [ 11665, 13369, 9 ], [ 13369, 15029, 10 ], [ 15029, 16760, 11 ], [ 16760, 18821, 12 ], [ 18821, 22614, 13 ], [ 22614, 26172, 14 ], [ 26172, 29801, 15 ], [ 29801, 33699, 16 ], [ 33699, 37311, 17 ], [ 37311, 40991, 18 ] ], "primary_language": [ "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en" ], "is_rotation_valid": [ true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true ], "rotation_correction": [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], "is_table": [ false, false, false, false, false, false, false, false, false, false, false, false, true, true, true, true, true, true ], "is_diagram": [ false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false ] }, "jstor_metadata": { "identifier": "jstor-107810", "title": "On the Mathematical Theory of Suspension Bridges, with Tables for Facilitating Their Construction", "authors": "Davies Gilbert", "year": 1826, "volume": "116", "journal": "Philosophical Transactions of the Royal Society of London", "page_count": 18, "jstor_url": "https://www.jstor.org/stable/107810" } }