SIE B. C. BRODIE ON THE CALCULUS OF CHEMICAL OPERATIONS. 113 
ffay)=f{a, b)+f x (a, b)(x-a)-\-f QA {a, b){y-b), 
||/ 2 MX*-«) 2 =o, 
b)(x-a)(y-b) = 0, 
p-/s(®> ^)(^-«) 3 = 0, 
J)(^— «) 2 (2'~ J )=°> 
&)(#— a )(^“ 5 ) 2=0 > 
[|/o.3(«,^)(y-J) 3 = 0, 
b)(x-a) m (y-b) n = 0. 
If /(#,«/) be a chemical equation, so that 
M 30=0, 
f(a , 3) is also a chemical equation, so that 
f(a,b) = 0; 
and, moreover, observing that 
/ (a,b)(x-a)=f(x,b)-f(a,b), 
fo,(a, b)(y-b)=f(a,y)-f(a,b), 
f(cc, b) and f(a , ?/) also being chemical equations, we have 
f {a,b)(x-a)= 0, 
/o.i(«, fl)(y-0}=°* 
In the two last equations, taken together with the equation 3)(#— a) 2 =0 and 
the succeeding equations of the system, the theoretical analysis of the event f(x, y )= 0 
is effected, since these equations are collectively identical, with the equation/^, y)~ 0, 
and in all respects adequately represent that equation. 
By the application of these principles we arrive at the following rule for the develop- 
ment of a chemical function^#, y, z, v, w , . . .), which satisfies the congruence 
f(x,y,z,v,w , . . .) =f(a, b,c, d,e,. ..), mod (x—a) mod (y— b) mod(s— c) mod (v—d) 
Develop by Taylor’s theorem f[x- \-Ax, y + Ay, z-\-Az, v-\-Av, . . .) in ascending 
powers of Ax, Ay, A z, Av, Aw, . . . Substitute in the development thus effected for 
