398 JACKSON. 



be solved explicitly, having infinitely many roots given by the relation 



PnV (W2 - wi) = log (^- ^^^j + 2 rnri, 



where the first term on the right is understood to refer to a particular 

 determination of the logarithm, and n is any integer. It appears from 

 the determinant expressions for 5i and 59 that each of these numbers is 

 either the conjugate or the negative of the conjugate of the other; 

 consequently they have the same absolute value, SiWi^" and 52^2*" 

 likewise, and the above logarithm is a pure imaginary. Hence, noting 

 that 



W2 — Wi = 2i sm -•, 



V 



we may write 



Pn 



sin (jr/vy 



where ^uq is a real constant. These roots are all of the first order, for 

 the derivative of ZX2 never vanishes. The passage from the roots of 

 Ai to those of Ai is effected by reasoning of the same sort as that used 

 in connection with the simple case of the third order. The conclusion 

 is as follows: 



The equation A = has an infinite number of roots, which can be 

 written in the form 



(^^^ "'= sin(x/.) ' n = g,g+l,g + 2,..., 



where g is some integer, positive, negative, or zero,^^ and ^'^ 



(21) ^'"^ en = 0. 



With the exception of a finite number at most, the roots are of the 

 first order. 



Of course it still remains to consider the possibility of the existence 

 of roots in the region Tov^i. But an argument analogous to that just 

 completed shows that, at least from a certain point on, this region can 



22 Of course it would be possible to adjust mo so as to make g equal to zero. 



23 For the sake of convenience, a definition of «„ is given here which does not 

 exactl}' correspond with that used in the previous special discussion. 



