558 PROCEEDINGS OF THE AMERICAN ACADEMY. 



One further remark comes in appropriately at this point. The 

 definition of Y (.r) and Y-{.v) ensured that | Y- (x) \ and | Y+ (.r) | do 

 not vanish to the left or right of the axis of imaginaries ^especti^'ely, 

 save possibly for x = oo . Hence | Y~{x) \ and | Y+{x) \ do not vanish 

 in these left or right half planes respectively. 



We may now enter upon a series of modifications of Y(x) which will 

 preserve the above stated properties and secure in addition that the 

 determinant of the leading coefficients of S{x) is not zero. To this 

 end we write the matrix S{x) in the form 



.T^ (p^e-'^yx"^ (an + ^.' + ...),.. ., x'^' {pne-'^)' x^n (ci,,^ + ^' + . . .^i 

 j 



a-'- ip,e-'^yx^^Ua+ ^+ ...),.. ., x'^^{pne->^)\fn (a,, +y + • • .)i 



and carry out reductions parallel to those given in § 11. The same 

 reductions are supposed to be simultaneously effected upon i ~(.i") and 

 F+(a:) (compare § 15). This set of reductions will terminate, since 

 1 »S(.r) I does not vanish identically, and, when it does, }"~(.i') and Y+{x) 

 will have the desired additional property. 

 Consider now the matrix 



Q (.r) = }'- {x + 1) [F- (.r)]-i = Y- (x + 1) [Y- (.r)]-'. 



These two forms for Q{z) are equal in ^■irtuc of (53) and yield us at 

 once the asymptotic form of the elements of Q{x) in the complete 

 vicinity of x = oo as descending power series in x with leading term of 

 the ixth. degree at most.^^ Hence the elements of Q (x) are rational 

 in character at x =^ , with a pole of at most order ^t there. 



In the finite plane to the left or along the axis of imaginaries the 

 first expression for Q{x) shows that the elements of Q{x) are analytic 

 without exception. It will be recalled that 1 ~(.r) is a matrix of entire 

 function of determinant not zero in the left half plane inclusive of the 

 axis of imaginaries. On the other hand the elements of Q{x) are ana- 

 lytic to the right of the axis of imaginaries, as the second form shows. 



Accordingly Q (.r) is a matrix of polynomials of degree at most /x 

 and r~(.v)> Y+{x) are solutions of the rational difference system (44). 



Finally it may be observed that in case | P{x) \ = along the axis 

 of imaginaries, a parallel line may be used to take the same role; or 



28 Cf. I, § 7. 



