410 JACKSON. 



characteristic values occur on the ray arg p = t^Iv, which bounds So, 

 the corresponding values on the ray arg p = — tt/v may be left out of 

 account. It will be convenient now to have a definite understanding 

 as to four of the subscripts of the roots uy, we shall write 



iri W 3)r» Sttx 



Wi = e " , 102 = 6", wz = e " , Wi= e " , 



so that the first four subscripts are assigned to the four right-hand 

 vertices of the j'-sided regular polygon w^hich represents the v roots. 

 If the polygon is rotated in the positive direction through an angle 

 6 between and ir/v, the vertices representing the numbers wie*', .... 

 ?f"4e^^ will be those furthest to the right, in the order of decreasing 

 abscissas. The order of the remaining subscripts is immaterial. 

 The quantities Wg (yi) have the asymptotic expressions: 



TVs (yt) = (pwO*« e ""^t^ [1] + as {pwiYil], s=v-1,p, 



where Ks is the order of the highest derivative whose value at the point 

 actually enters into the s-th. condition, for the last two values of s, 

 and as is the corresponding coefficient, or is zero if the condition does 

 not involve the point at all. Let the factor p ^s be divided from the 

 s-th row of the determinant of these quantities, s = 1, 2, . . ., v, and 

 in the last two rows let js he \\Titten for Ks — kg- Then the condition 

 for a characteristic value is expressed by the vanishing of a determi- 

 nant Ai of the following form : ^* 



V'[l] w-z'^'ll] ••• t^>[l] 



+ a.-i p'^f-i wi""-' [1] ) ( + a,_i p-'v-i w-i'v-' [1] 



f W/"-! e""'"'' [1] 1 



\ + a,_i p^j.-i MJ/'"-! [1] j 

 w/"" e""'"^ [1] ) j 10'^^ C'"-'^ [1] 



+ a, pT" Wi"" [1] j 1 + a, py w-^" [1] 



wj"" e''"''''^[l] 

 + a. pyw/il] 



34 This does not correspond precisely to the determinant Ai of the preceding 

 section, as no exponential factor has been divided from the last rows. 



