350 PROCEEDINGS OF THE AMERICAN ACADEMY. 



exponents of the set (23) are unity, and in this case the restriction drops 

 out also. 



The lower limits of integration in (22) will be determined as follows : 



(24) 



I. R r k > R r K c h i = c 



II. Rr, = R r K 



( e k> i == I < L — e K — e K + A 

 (c kii = e 1>L 



III. Rr k< Rr K c ki =0 



where c is a constant not zero, which will be determined more closely 

 later (cf. p. 358). It will be proved in § 3, that even in the cases in 

 which the lower limit is zero, the integrals converge. 



When all these conditions have been satisfied, we build from the first 

 approximation and the successive corrections the n infinite series : 



q=co 



(25) **/= 2^*. 



g=o 



k = 1, 2, . . ! rn 



I = 1, 2, . . . € k 



' 



which will be proved in § 4 to converge and to form a solution of the 

 system of equations (16). 



It will be convenient to consider in place of the functions z k !q certain 

 new functions <J3 ktli9 , which will be defined by the following formulae: 



(26) 



when R r k ^ R r K , 

 z k,i, g : = * K $k,i. g \™ Rr i = Rr K , k dp k and I < L, 



or k = k and I < A, 



«k,i, g = * K (log x) l ~ L <\> kJ>q Rr k = Rr K , k ^ k and L < / < e k , 





= x* (log a;)" ^ lq X <l^e K 



For the case of q = 0, i. e., the first approximation, the <£'s are certain 

 constants (cf. (11)); for all other values of q the formulae (26) define 

 them as continuous functions of x so long as x is not zero ; and we shall 

 see later (Proof of Convergence, p. 358 et seq.) that each one approaches 

 zero when x approaches zero. We shall therefore define each <$> k ,i, q -> 

 when q > 0, as zero for x = ; and with this definition they will be 

 continuous functions of x in the whole interval Ob. 



These «£'s can be computed from the following recurrent formulae, 

 which are easily obtained from (21), (22), (24), and (26) : 



