Kkllehek — A 3- Dimensional Complex Variable. 91 



unity, and in that case the point u' lies inside the tetrahedron whose vertices 

 are the points u, 6u, 6 ! u and the origin. Hence, if we denote by A, B, C the 

 points u, Ou, and Bhi respectively, and by the origin, and if we rotate the 

 line OA about the line x = y = z through 120° so that u becomes Bit, and then 

 again through 120° so that 6u becomes 9"-io, the point u' will lie inside the 

 tetrahedron OABC. It follows that if we describe the right cone whose 

 vertex is the origin and axis the line x = y -= z, and the tangent of whose 

 semi vertical angle is twice the tangent of the angle made by the line joining 

 u' to the origin with the line x = y = z, then the infinite series is convergent 



when the point u lies outside this cone and, since mod. u = — and 



p + q + r 



p + q + r is less than unity, outside the space containing the origin and 

 bounded by the planes mod. u - mod. u' = 0, and mod. u + mod. u' = 0. 



It does not follow, however, that the series in par. VI, and that which has 

 just been considered, are not convergent in other cases. 



