But, by Art. (45), when we differentiate considering y. constant,

z being a function of x and y by virtue of the equation

+ y log, x.cos (xy) dx + log, x.x22. {z + xy cos(xy)} dy.

Ex. 15. Given that

(xy)}

CHAPTER III.

SUCCESSIVE DIFFERENTIATION.

Theory of the Independent Variable.

46. Let (x, y) = 0, where (x, y) denotes any function of x and y whatever. When for x we substitute the successive values x + x, x + 28x, x + 38x,... .let the corresponding values of y be y1, y2, Y.,... .Then, Sy denoting the increment of y due to the increment dx of x, we have

y2

hence, putting y + dy for y and y, for y, in this equation, which corresponds to the change in the equation due to giving x another increment Sx,

or, putting Sdy = 8y, as an abbreviation of notation,

Similarly, x receiving a third increment S,

Y1 = y + dy + 28 (y + dy) + 82 (y + Sy)

Proceeding in the same way, we shall finally get, the law of the coefficients being evidently the same as in the binomial theorem,

Thus we see that as a keeps increasing by equal increments dx, y generally increases by unequal increments: in fact the increment of y, corresponding to an increment dx of x, is dy, and, for an increment ndx of x, it is not ndy, but