5G6 PROCEEDINGS OF THE AMERICAN ACADEMY. 



wise any ith column of Y^ (.r) is only determined up to a constant 

 factor hi. This fact appears from (58). The efi'ect is to allow one 

 to replace c^- in py(.r) by ffij/hi, and thus vary at will 2 n — 1 of the 

 characteristic constants c,,-. There remain then essentially onlv 



2?? + n- (m + 1) - n^ - (2n — 1) or n- fx -\- 1 



characteristic constants. It is readily verified that these constants 

 are all invariant under a transformation Y{.r) = Cy{x), where C is 

 an arbitrary matrix of constants. 



But the equation (54) invoh'es n-(fj. + 1) arbitrary coefficients in 

 Qix) of which there are n~ (ju + 1) — {n~ — 1) or n-fj. + 1 invariants 

 under the same linear transformation. Hence we have found as many 

 invariants for the g-difference system as invariant characteristic 

 numbers for the solutions. 



We are thus led to formulate the following problem: To construct 

 a ^-difference system (54), with coefficients of degree not greater than 

 /x in X, having any assigned set of characteristic constants. 



§ 21. Solufion of the Problem of § 20. 



Here also we shall employ the preliminary theorem, but the appli- 

 cation of it is even simpler than in the earlier cases. 



The conclusion that we shall derive is the following: There exists 

 a linear q-diffcrcncc system (54) with the mcdrix solutions Yq (x), Y^^ (x) 

 either possessing prescribed characteristic constants Pi, <Ji, Cij, a/''^^, . . .,aj''^' 

 or else constants pj, Cj + Ij, Cij, a/''-'^, . . . , aj:^'^\ tchere /i, . . . , /„ arc integers. 

 For an arbitrary loop about x = lohich cuts each spiral (70) 07ily once 

 and does not pass through a point \ P{.r) | = 0, there exist matrices 

 Fo (.r), Y^ {x) with the further property that \ Yq (a-) | 9^ tvithin or along 

 the loop while the elements of Y^{x) are analytic and \ Y^{.v) \ is not 

 zero without the loop. 



Let Ci be a specified loop of this description which may be taken to 

 be analytic. ^^ We may take the matri.x Ai (.r) of the preliminary 

 theorem to be 



T^{x)Pix)To-H^), 

 where ^ 



To{x) = {x'ibif), T^ (x) = q2^' '^ i^'^^Sii), 



since the elements of -4i(.r) are single- valued and analytic along Ci 

 by (60). 



36 It is only a question as to how the loop weaves among the zeros of |P(x)| . 



