Kellkher — A 3- Dimensional Complex Variable. 89 



that all the " A "s are fractions which have the denominator 



f i t 



X\ — X\ j 3/2 — X-i j *wj •* n 



%2 — X% y 3*3 — X$ ) ^1 -^1 



X n — X n , X\ — X\ , X-n-\ ~ %n-\ 



a circulant whose only real factor is S(x p - x t ') when n is odd and which has 

 the additional real factor 2 (- l) p (x p - * P ) w hen n is even. 

 Similarly, if the function to be integrated were 



i> («) 



(u - vf) [u - u") ...(«- «<«>) 

 the path of integration must in general avoid the parallel planes 



S(«, - a,/) = 0, S(x t - *i") = ... 2(0,-! - x,W) = 0, 

 when n is odd, and in addition the planes 

 2 (- 1)p(x p - x/) = 0, 2 (- l)*^ - V) = . . . S (- 1Y (x p - x p W) = 



when n is even. 



In the case of three or any odd number of dimensions we may express the 

 result by saying that the path of integration must pass through no point 

 whose modulus is equal to that of any one of the points ii, v.", . . . »W). 



V. If we consider the case of three dimensions, and use rectangular 

 coordinates, we see that the point Qv, is the point derived from u by rotating 

 the vector from the origin to the point u through 120" about the line se = y = z. 

 For multiplication by ti of x + yd + z8~ gives us z + xQ + y0~ ; and if we call 

 the points u and 6u A and B respectively the projection of each on the line 

 x = y = zis the point 



Lm * + y + * (1 + e + n 



Hence from the equation 



AB- = LA' + LR- - 2LA . LB cos ALB 

 we find cos ALB = - J, 



.-. ALB = 120 c . 

 When we consider the ease of n dimensions and regard x u x 2 , . . . x n as the 



