108 BIRKHOFF AND LANGER. 



b 



■Z {1) ( R(x)l 



Multiplying this equation by Y w (x)- on the left it becomes 



b 



(0/ w n _ / v(0/^..y(0 







I 



-f' 



qr'(x)-(l) = J Y w (z)'-Z w (t) R(t) F(i)-dt, 



a 



which in view of (85) and (89) reduces to 



(94) c t Y (l) {x)- = - J G (l \x, t) R(t) F(t)-dt. 



a 



This expression for the l' term of the formal development was de- 

 duced upon the hypothesis that Xj is a simple characteristic value. 

 If at Xj a number of characteristic values \ h , \ k ,. . /X tll , coincide, we 

 shall define the term of the formal series which corresponds to this 

 value of X to be 



6 



- fG w (.r,l)R(t)F(t)-di. 



a 



In every case then we have a formal series which is completely de- 

 termined and of which the I th term is given by formula (94). It is to 

 be noted that G (x, 1) for the case in which X/ is not a simple charac- 

 teristic value is not given by the left-hand side of (94). It must be 

 computed for every such case individually. 



Now let pi be any contour in the X plane which surrounds X/ but no 

 other characteristic values. Then by use of the relation 



we have from (94) 



,X) f /X= G ( '\x,t) 



^7 (0 (.r)- = - -J-. / I G(x,t,\)d\R(t)F(t)-dt, 



a p t 



or, upon interchanging the order of integration and choosing a contour 

 C m which encloses the characteristic values Xi, X 2 , . . .X m and no others, 



6 



(95) 



£c, Y (k \.r)- = - -^ / / G(x, t, X) R(f) F(t)-di dX 



m 



