80 Mr Bromwich, Theorems on Matrices and Bilinear Forms. 



can be determined ; the method is, also, capable of being extended 

 to functions, which are not rational and algebraic. 



Suppose f{r) any rational function of r and then consider the 

 value of the sum of Cauchy's residues for the function of r 



fir) yjrjr) -^ js) 



ty (r) r — s 



at the points r = a, b, c, We see at once that this sum is 



equal to 



fis) — I sum of residues of , , {; \ for \ 

 J w I i|r (r) (r - s) J 



\ the points r = s,r = a,b,c,... / 



Now considering all the residues of /(r)/[\|r (r) (r — s)] we know 

 by a theorem of Cauchy's that their complete sum is zero ; and in 

 addition to the points r = s, r = a, b, c, ... there are the poles of 

 fir) and possibly r = x . Thus the original sum of residues is 



(fir) \ 



sum of residues of , /. \ for \ 

 y (?-) (r — s) j 

 the poles of fir-) and r = oo / 



=f(s) + iris)gis), 



where g (s) has the property of not being infinite for any of the 



values s = a, b, c, . . . , for we suppose that r = a, b, c, ... are not 



poles of fir) otherwise the function of the given bilinear form 



would have no meaning. It will be recognized that g is) is the 



fis) 

 sum of those partial fractions in , . , which are not infinite for 

 * ^is) _ 



s = a, b, c, ... ; so that g iA) has a meaning. 



Now write s = A in the result last obtained (which we may do, 



for the equation is rational on each side) and then, since ty iA) = 



and since g iA) has a meaning, 



fiA) = sum of residues of irE — A) -1 fir) for 



the points r =a, b, c, .... 



We can write this in a different form, for we have an equation 

 of the type 



irE -A)~ l 



Ai -n.o 



= -f J -f- . 



r — a (r — a) 2 



+ A ° 





B 1 B 2 



r—b (r — b) 2 



+ B > 



(r - by 





+ ..., 





