any value we please in (3), we may replace b by x, and write

x (x, c)

=

dy (x, c)
dc

+ 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)