108 SIE B. C. BEODIE ON THE CALCULUS OF CHEMICAL OPEEATIONS. 
Hence 
A 2 (#— a) 2 =0, 
and 
A„(x— a) n =0, 
f(a-\-x—a)=A 0 -\-A 1 (x—a). 
Since this last equation is always true, it is true when x—a ; putting x=a, 
f{a)= A 0 ; 
and also since the residue E is the value of f(x), when x=a. 
and 
To determine A, we have 
A 0 =E, 
f(x)=f(a)+A 1 (x-a). 
Aj— 
This equation again being always true is true when x=a ; but the limit of the value of 
f{x) —f(a) , 
x—a ’ 
when x=a, is where f\(a) is the first derived function oif(x), that is hi which 
a is substituted for x. We have therefore for the value of A n 
A i=/x(«), 
and 
f(x) =f(a ) +f\(a)(x - a). 
Similarly, in the case of the congruence 
fix, y)=f(a, b ), mod (x-a) mod ( y-b ), 
we have, regarding y as a constant and developing by the above theorem, 
/O, y)=f(a> y)+fi{a> y){x- a l 
and also 
£)+/(«> b){y—V)r 
/(«» y)=f i(«> £)+/i.i(«> 
whence, substituting these values for/j^, y) and/i(«, y), we have 
y)=f(a , b)+f uo (a , 5)(^-«)+/o t>)(y-t>)+fi. i(«, 
and 
/0> y)=/(®, s)+/,.;( a , &)(*-«)+/..,(«, j)(y-5). 
