112 
SIR B. C. BRODIE ON THE CALCULUS OE CHEMICAL OPERATIONS. 
and generally 
and let 
f. 
f. 
Jm.n{X,y)— d m X ' d n y > 
/„..(«> b)=f m . n {x,y). 
in which « is substituted for x, and b for y. 
Now in f(x,y) let x and y respectively vary, so that x becomes x-\-Ax and y becomes 
y-\-Ay; we then have, by Taylor’s theorem, 
f{x+Ax, y+ A y)=f(x, y) +f(x, y)Ax+f 0 . ,{x, y) Ay+ ~f 2 (x, y)Ax 2 
+fi.i(^y)AxAy+ T ^f 0 . 2 (x,y)Ay 2 +^f 3 (x i y)Ax 3 +^f 2 . l (^,y)Ax 2 Ay 
+ j^/i . 2 (^, y)AxAy 2 +^f 0 ' 3 (x : y)Ay 3 . . . +~^f m _ n (x,y)Ax m Ay n i 
whence, assuming as before Ax to be the variation which x undergoes when x is sub- 
stuted for a, and Ay the variation which y undergoes when y is substituted for b, we 
have 
A x=a—x, 
A y—b—y. 
Substituting these values in the above development for Ax and Ay, and again changing 
in the result a into x and x into a, and b into y and y into b, we have 
/(tf> S') ==/(«> ^)+./iK a)+/ 0 .i(a, b)(y- b) +^f 2 {a,b){x-df+f 1 . ^b^x-a^y-b) 
+pr/o. 2 («, b){y-b) 2 +^flci, b){x-a) 3 -\-^~f 2 , x {a, b){x-af(y-b) 
+j \f >2 («, b){x-a){y-b) 2 +^f 0 ' Z {a, b){y-bf . . . +^f m . n (a, b){x-a) m {y-b)\ 
On the principles laid down in the last case, this equation may be resolved into the 
following system of equations, by which it is adequately represented : — 
