Electric Waves round a Large Sphere. 285 



I£ Y n =[d n Y/dco n ] 



w=o 



then by Maclaurin's theorem 



V = V + V l0 )+^|o> s + ..., . . . (121) 



and 



$e~ tzv «/2z= (T + f" } dcoYe 1 ^ 



Now by integration in the plane of a complex variable 

 round a contour consisting of two lines at an angle \ir 

 through the point h on the real axis, and an arc at infinity, 

 it is readily shown that 



/loo 



= e *P+b* | dr Y(-S-re 4 ) e e(-r*+2ir8e T ). 



Jo 



The latter integral may be evaluated in a series of powers 

 of z~ Y by parts. The leading term of the series becomes 



V(- 8)e tzS2 /2iz$ (122) 



But V( — S) is the value of V when &>= — S, that is, the value 

 of V when x = e. Thus 



V(-S) = [U«/«']. 



x—e 



Consider then the case in which the exact formula (112) 

 is used. The leading term of U is of the same order as a, 

 and when x; = e, this order is, since m — i, zero in z. More- 

 over, 8 is not small, and therefore V( — 8)/2iz8 has an order 

 z~ l at most. On the contrary, in the integral 



C de>Y Q e 1z<o * = Y ( dve lzl 



Jo Jo 



V has an order s, being the value at the zero point where 

 its factor m has this order. The resulting order of the 

 integral is at. Thus Y( — 8)j2iz8 may be neglected, with an 

 error at most of order jsr — t compared with the main order 

 retained. This will be found sufficient to justify its neglect 

 in a determination of the second approximation, and there- 

 fore the matter is not investigated further.' But in reality. 



