366 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Now by T. no logarithms appear explicitly in the terms (72), for the 

 only case in which they could occur would be for X = 1, but by (71) 

 such terms do not appear. We can then write each term of (72) 



73) 0« f x«T f, **2, A = C K , K x r *-**<f>&; 



and by (63) the limit of each is zero except in the one case : 



(74) limit C k 2 x~ Tk z\ I = limit G k 2 <£*; ■ = C k2 . 



x=0 x=0 



We have then, reasoning as before, and including the previous result, 



(75) GV= ^,2 = Rr K = Rr v 

 Now the same reasoning can be applied until we have finally 



(76) Oc,x = 0; Rr K = Rr x , 



X > 1 



and there are left in (65) only those solutions for which Rr K > Rr x . 

 Having thus disposed of all the r'a in (66) such that Rr K z=Rrj,we 

 consider the next set of r's whose real parts are all equal and as small as 

 any other in the new set of r's, and show, in exactly the same way, that 

 the corresponding constants, C K> \ are zero. Continuing in this way, we 

 finally reach the result that all the constants in (65) are zero, and that 

 the assumption made as to the dependence of the n solutions leads to a 

 contradiction. The n solutions are therefore linearly independent. 



§ 6 



Rr.TURN FROM THE CANONICAL SYSTEM TO THE ORIGINAL SYSTEM. 



We will now return to the original system of equations (1). Each 

 solution of the canonical system (16) determines a solution of the original 

 system (1) on account of the relation (10). We have then the following 

 n solutions of (1) : 



k=m i=e k (t = 1, 2, .. . n " 



(77) #*= 2 2 a^S 



k=l 1=1 



k = 1, 2, . . . m 



X = 1, 2, . . . e K J 



Now these n solutions are linearly independent, for the z solutions are 

 linearly independent, and the determinant of the constants A iihl is not 

 zero. 



