386 JACKSON. 



closely correspond mth those of (5). Let us show that this is the 

 case,^ 



Let € be an arbitrarily small positive quantity, and let R be the 

 region obtained by removing from the whole p-plane the interiors of all 

 circles of radius e with centres at the points p'n. If the left-hand mem- 

 ber of (3), minus the first term, is denoted by ^ (p), then, in the identity 



^p (p) = coe-'-"'^ (1 4- ue^^p''^), 



the last factor on the right is a function of p having the period 2/ V3. 

 It vanishes only at the points p'n, and becomes infinite uniformly or 

 approaches uniformly the limit 1 as the imaginary part of p becomes 

 negatively or positively infinite. In the region R, consequently, its 

 absolute value has a positive minimum 8. The absolute value of the 

 parenthesis in the alternative identity 



<p (p) = co^e-"'"'^ (1 + a;2e->/3pW) 



has the same minimum 5, for the two parentheses take on con- 

 jugate imaginary values for conjugate imaginary values of p. Let 

 a circle be described with centre at the origin and radius so large that, 

 in the part of the sector (4) which lies outside it, ] e-""^ I < P; and 

 suppose p restricted for the moment to this part of the sector. If 

 p is exterior to all the little circles, (p (p) -f e-""^ is surely not zero, 

 because of the inequalities 



(6) k(p)|>5, k-'"^|<i5, 



the first of which is a consequence of the definition of d and the fact 

 that at least one of the quantities e~"P'r, e~"^p'^, is in absolute value 

 greater than 1. On the other hand, if p describes the circumference 

 of one of the little circles in the positive sense, the argument of (p (p) 

 increases by 'Iw in a single circuit, since (p has just one simple ^ root 

 inside, and it follows from the inequalities (6) that the argument of 

 <p (p) -f- e-p"" must do likewise. Hence (3) has just one root in each 

 of the little circles. That these roots are actually and not merely 

 asymptotically real, appears from the fact that complex roots of (3) 



8 The discussion of this point is put in a form adapted to the demonstration 

 of Theorem I; cf. Birkhoff II, pp. 384-386; Tamarkine I, pp. 353-358. It 

 yields more than is needed for the proof of Theorem II, and may be replaced 

 by a simpler argument, as far as that theorem is concerned; see below. 



9 It is immediately verified that ^(p) and its derivative can not vanish simul- 

 taneously. The same is true of ,p{p) + e~p^ for p ?^ 0, so that aU the roots of 

 (3) which give characteristic values, even those in the finite region which has 

 been temporarily left out of account, are of the first order. 



