Examples of the Processes of the Differential and Integral Calculus (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1841 Excerpt: ...-2); therefore if we assume. 2lTZ 2lT- x = a sin-----, y = a cos-----, A' h the preceding expressions will vanish, and therefore the line determined by these equations, and the equation to the surface is a locus of singular points. This line is the intersection of the surface by the cylinder C 2 - a + y = a, and is evidently the generating helix. Since in the equation to the surface a's + y2 can never be less than a1, it appears that no part of the surface lies within the helix, which is therefore truly an edge of regression. On proceeding to the second differential coefficients, and substituting in them the critical values of x and y we find, retaining only the terms which become infinite from involving (a;2 + y2-a?)l in the denominator, (-) =-2 sm----cos----, (i) = 2 sin----cos----, (w) = 0, h h h h, '2ir 2ir%, . 27r. 2tt, . 27r 2?r# (u )=--r-acos--r--, (v )=---a sm----, (w )=snr--cos-----; nana h h so that the equation to the locus of the tangent lines is a2 (y'a-w'2) asy + x'y' (jx2-y2) + 2w--%' (w'x + y'y) = 0, where the accentuated letters are the current co-ordinates of the tangents, and the unaccentuated the undetermined coordinates of the point of contact. This equation may be decomposed into two factors, y w--no y + 2 7r--ss = 0, h x'x + y'y = 0, which are the equations to two planes. Umbilici. These are points at which the two principal radii of curvature are equal. The conditions for determining them are 1 + p2 _ pq _ 1 + q2 r s t (6) In the ellipsoid 4 2 c x c y Hence we have a1 + 4, .i? aa a'2 + 4i/2 2 a.0 2 a' In order that these equations may hold we must have either v = 0, or y = 0. Taking the former we find 2j/2 a a.., a-a' a 2 2 4 Now if a a the value of y is possible, and there are two umbilici, ...