48 -^rt. 7.— K. Terazawa : 



^•;_;„. sin/l ,;_ , ^^_i 1 



,/' 



f 



~^-''coü?uU = 



l + x' 



the evaluation of the function Il,„ (:r) can l»e undertaken. A httle 

 calculation will give us 



IJ^^(x) = - — •|a: + tan"^a: — a;-taii"^ > , 



4 2 1 X ) 



ßi(x) = 1 — j:tan~^ , 



X 



(L{x) = tan-'--- —*-- , 

 X 1+a?" 



Î2,{x) ^ 



(l+x')- 

 and in general 



i2,„U) = (-1)"-' -^{-^1 



{l + xj 



Thus the integrals on the left liand side of (103) can be expanded 

 in accending power series of rja which probably converge for 

 limited values of r if the value of z is fixed. These series and 

 those found in (lOO) ha\'e a common region in wliich the}^ are 

 both convergent and tlierefore they must be congruent to each 

 other in that region. On the proof of tliis proposition we shall 

 not enter; but we shall find the region of convergency of these 

 latter series at the boundry. Let us take the first series of (103). 

 Expand the function //„+i(-^) for n^l into a power series of zja, 

 supposing zja to be sufficiently small, then tlie first term of it wiU 

 be {-l%i{%i-'2) !. Thus if we put ,? = in the first series of (103), 

 its general term will tlien be 



%i{-2n-'2)\ / r ^-"+^ 

 n\ (n+1)! 2-"+^ \ a 



