any value we please in (3), we may replace b by x, and write x (x, c) = dy (x, c) + C .(4). This equation may be applied to find x(x, c); as the constant may be introduced if required, we may dispense with writing it, and put (4) in the form 216. Required the differential coefficient of [" S* $ (x, c) di a dx with respect to c when both b and a are functions of c. Denote the integral by u; then consists of three terms, du dc one arising from the fact that p(x, c) contains c, one from the fact that b contains c, and one from the fact that a contains c. 218. The following geometrical illustration may be given of Art. 216. Let y(x, c) be the equation to the curve APQ, and y=4(x, c + Ac) the equation to the curve A'P'Q'. Let OM=α, ON=b, MM' = ▲a, NN' Ab. H Then u denotes the area PMNQ, and u+ Au denotes the area P'M'N'Q'. Hence Au = P'pqQ+ QNN'q — PMM'p, It may easily be seen that the limit of the first term is the (x, c + Ac) (x, c) Δε φ Ac' second term is the limit of, (b, c) of the third term is the limit of result of Art. 216. 219. Example. Find a curve such that the area between the curve, the axis of x, and any ordinate, shall bear a constant ratio to the rectangle contained by that ordinate and the corresponding abscissa. Suppose (x) the ordinate of the curve to the abscissa x; then $ (2) da expresses the area between the curve, the hence we may differentiate with respect to c; thus By integration log (c) = (n − 1) log c+constant; thus and $ (c) = Acr ̄1, ¤ (x) = Ax2¬1, which determines the required curve. 220. Find the form of (x), so that for all values of c [° x {$ (x)}a dx = = [* {$ (x)}* dx. n where A is some constant; thus we have finally 2-n (x) = Ax2 (n−1) ̧ This is the solution of a problem in Analytical Statics, which may be enunciated thus: The distance of the centre of gravity of a segment of a solid of revolution from the 1 vertex is always th part of the height of the segment; find n the generating curve. The required equation is y = (x). 221. Find the form of $ (∞) so that the integral may be independent of c. Denote the integral by u, and suppose x = cz; thus = √ √ep (cz) dz = 0 √(1-2) (a) dx √(c-x) |