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

OF SEVERAL VARtIABLES.

and theni from (2)
2 X+a 2 (ay (M ay), d2y
y-aie2a + 2Y +( x 2=
dy2 daX x) y
which Igives,d; and so we may find y
(See Appendi x F.)
tt 109. If y be a function of u and v, u and v being Successive
differentiaindependant variables, then supposing du and dv constant (see tion of a
94), we have, as i (107) ucin
two independant
dy = d,,y du + dvy. dv, variables~
d2y = d2y-. dU2 +2d~d~y -du dv + d,2y. dV2,
d3 y =d4'y. du' + 3d~dvy. du2 dv + 3 d~d2Y. du dv2 + d'y. dv3,
We perceive that the coefficients of the terms in the second
members of these equations are respectively those in the expansions of 1 + z, (I + _2), (I + Z)3
Le, A,~ B, C... be the coefficients in the expansio
of (1 + z~n, and let us assume that
d~y~~y~Un+Ad' d y.dull. dv+Bd~2d~y. d un -dva +
then dn+ y-=d,(dny) -du +dv (dny).dv
-dn+1y. du"+' + Add~y. du"'dv + Bd~'diy dl dv
Now 1, A+ i, B +A, &c. we know to he the coefficients
in (1 + z)n+' expanded: hence it appears that if the law we
have assumed be true for n it is true also for n + 1; but
we know it to be true for 1, 2, 3;- therefore it is true in
general. We have therefore
nn
dny= d Y. du" + - d~'ld,,y.du,,' dv
n (n -1) d-2d2'y. du2dV2
+ d