An elementary course of infinitesimal calculus . Fig. 117. Ex. 2. In the ellipse (or hyperbola), if QR, drawn parallelto the tangent at either extremity of the diameter PGP, meetthis diameter in V, we have Q7^\P7. VP = GD^: GP^,. Fig. 118. where CD is the semi-diameter conjugate to GP. Hence, for thechord of curvature (q) through the centre, g = lim-p^ = hm^,. FP =2 ^ (4). 153] CURVATURE. 405 If GZ be the perpendicular from the centre on the tangent atP, and B the angle which GP makes with the normal, we havecos B = GZjGP, and therefore p = iqseoe = -^ (5), in agreement with Art. 151 (17). Aga
An elementary course of infinitesimal calculus . Fig. 117. Ex. 2. In the ellipse (or hyperbola), if QR, drawn parallelto the tangent at either extremity of the diameter PGP, meetthis diameter in V, we have Q7^\P7. VP = GD^: GP^,. Fig. 118. where CD is the semi-diameter conjugate to GP. Hence, for thechord of curvature (q) through the centre, g = lim-p^ = hm^,. FP =2 ^ (4). 153] CURVATURE. 405 If GZ be the perpendicular from the centre on the tangent atP, and B the angle which GP makes with the normal, we havecos B = GZjGP, and therefore p = iqseoe = -^ (5), in agreement with Art. 151 (17). Again, if 0 be the inclination of either focal distance to thenormal at P, it is known that cos 0 = GZjGA, where A is an ex-tremity of the major axis. The chord of curvature ((j) througheither focus is therefore given by g = 2pcose = 2^ (6). Ex. 3. To find the radius of curvature (po) at the vertex ofthe cycloid x = a{6 + sin 6), y = a{\ —cos6) (7). We have ^ la ? a\i i ? -itn /t sin ^N /sin \6\^2^ = a (6 + sin fl)^ - 4 sin^e = a (^1 +-^j - (^-^ j , a?whence p„ = lim9=o 9-= 4a (8). Newtons method, combined with the result of Art. 66, 2°leads to a general formula for the chord of curvature parallelto the axis of y, a
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