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

SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES. 63

d dy\ _ dy d"c
d dyv dy d2A, since dy is now constant;
d dy
dc~~v d" x 3 d', v d"x
d i dyv~ dx4
d~~v d X 3 dw x
Hence u= (3d"x + ) d2 - + - 3
divdy dxdy dv dy' dX 2dy4
dV d cv dy I eta? d3
dxdy divdy' li \dy2 dyv
in which expression y is the independant variable.
(For other examples see Appendix E).
The following articles have reference to the successive
differentiation of functions of several variables.
tt 104. Suppose that y is some function of u and v. successive
u and v beino functions of v; then by Rule XV. difrentiaf(utv), uV
dy = d,,y du + d'y dv; being functions of X.
therefore differentiating again
d2y = d (d, ydu) + d (dv ydv).
Now d (d y du) = d (dy). du + du y. d2 U
{d,, (d, y). du + d, (d, y). dv du + du ydlu., by Rule XV.
hence assuming du2y to denote ddy, we have
d (day du) = dly. duc2 + dd,, y. dudv + dyd2 u... (1);
and similarly we have
d(d~y dv) = d~y. dv2 + d,,dy. dudv + dyd~v... (2);
and adding (1) and (2), we obtain an expression for d2y.
This expression becomes somewhat simplified by means of the
following article.