. An elementary treatise on the differential and integral calculus. in practice, n istaken as , and for still ruder approximations as 3-J-. 175. The Ellipse.—From y2 = (1—e2) (a2—x2)9 we have &=_(l_^5=_?vL dx y A/a2 — x2 To find the length of a quadrant, we must integrate be-tween the limits 0 and a; hence, ^ v7-* RECTIFICATION OF THE CYCLOID. 351 This integration cannot be effected in finite terms, butmay be obtained by series. Put =z\ then dx = adz. When xa a, z = l, and when x = 0, z — 0 ; therefore the above integral becomes e2*2) efe Vi —z2 i a _ ±1 4 _ l^i 6 \ 4e * V
. An elementary treatise on the differential and integral calculus. in practice, n istaken as , and for still ruder approximations as 3-J-. 175. The Ellipse.—From y2 = (1—e2) (a2—x2)9 we have &=_(l_^5=_?vL dx y A/a2 — x2 To find the length of a quadrant, we must integrate be-tween the limits 0 and a; hence, ^ v7-* RECTIFICATION OF THE CYCLOID. 351 This integration cannot be effected in finite terms, butmay be obtained by series. Put =z\ then dx = adz. When xa a, z = l, and when x = 0, z — 0 ; therefore the above integral becomes e2*2) efe Vi —z2 i a _ ±1 4 _ l^i 6 \ 4e * V* (by Ex. 17, Art. 170), which is the length of a quadrant ofthe ellipse whose semi-major axis is a and eccentricity e. y 176. The Cycloid.—From x = rvers-1^ — \/2ry—y2, we have dx dV V%ry — y2 s = v 2r I (2r — y)~* dyJo [? 2 (2r) * (2r - y)* 2r = 4r, which is | the cycloidal arc; vhence the whole arc of the cy-cloid is 8r or 4 times the diam-eter of the generating circle. If we integrate the above ex-pression between y and 2r, we get. Fig. 44.« = V2r / (2r - y)^ dy = 2 (2r)* (2r — y)* = 2V2r(2r — y) — arc BP. But BD = a/BAxBC = V2r(2r— y); .-. arc BP = 2 times chord BD.* * This rectification was discovered by Wren. See Gregorys Examples, p. 421. 352 INVOLUTE OF A CIRCLE. 177. The Catenary.—A catenary is the curve assumed by a perfectly flexible string, whenits ends are fastened at two points,A and B, nearer together than thelength of the string. Its equation is y = £(«•+.-:-).
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