An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

66

6SCCESSIVE DIFFERENTIATION OF FUNCTIONS

dd,y = 1; and therefore we have
d`2y = 2du2+ 2dudv-t 2dV2 + (Qu + v) d 2U + (2v + u) d 2V.
Again d3y=O, d y=O, d d2y=o, d dvy= O;
and therefore we have
d3y = 3 (2du d'u + d'u dv + dv d'u + 2dv d'v)
+ (2u + v) d'U + (2+ u) d'v,
and so we may find d'y, &c.
The method here given of finding the successive differentials of a function of several variables is useful only for
general purposes. In particular cases we may always find
the successive differentials more readily by the common rules
of differentiation. (See Appendix F where examples of this
are given).
Successive j t 108. If U be a function of x and y, and we put
on ofan U = 0, then we have by Rule XVI.
equation
between
y and x. d U. dw + dy U. dy o,
and differentiating as in (107) we have, supposing dx constant,
d2U. dX'2 + 2d,dy U. dx dy+ d1 U. dy2 +d U.d y= 0,
dy. dly dlyy
and so we may go on, and by this means find dy d- ' d'y
&c. from the equation U= 0.
Ex. Let X'-3caxy + y3 = U= 0,
then d'U=3x-,3ay, d'U=6c, d U=3y'-3ax,
d U= 6y9 dvdyU 3 a: we have therefore
(X2 _ ay) dxv+ (y2 - ax) dy= 0...............v.. (1)
2xd 2-adxdy + 2ydyx+(y-a) dy = 0...(2),
dy x'-ay
from (1) we get
dx Y2 axa