90 Proceedings of the Royal Irish Academy. 



perpendicular distances from the point u = %i + x 2 9 + . . . x n 9 n ~ l to n planes 

 whose intersections are generators of a right circular cone, and such that each 

 makes equal angles with the two adjacent to it, multiplication by is 



equivalent to rotation through — about the axis of the cone. 



VI. If we write , in the form 



u - u 



1 





u" 



' 



■ —T„ + 



u 



( u 



U n 



where u is the 3-dimensional variable, we may show that the infinite series is 

 convergent in certain circumstances. For if we write 



it follows from par. I that if -^ is written in the form P„ + Q„9 + R„6' 



the modulus of P„ + Qd„ + R„6- is equal to (p + q + r)". Hence, when 



p, q, r are all positive and their sum less than unity, P u , Q„, B n are all 



positive, and each is less than (p + q + ?•)", and therefore the infinite series 



it" 

 2-- or 2P„ + 02Q„ + fl'SA. is less than (1 + + 1 ) 2{p + q + rf and is 



convergent. 



But since u = u'(p + qf) + rf) 1 ), it follows from par. V that if we consider 



the tetrahedron whose vertices are the points u', Ou', 0*-u', and the origin, then 



if p, q, r are all positive, and their sum less than unity, the point u is a point 



inside the tetrahedron. Moreover, since any point inside the tetrahedron 



may be written pu + qOu + r8*u where p, q, r are all positive, and their sum 



u" 

 less than unity, it follows that the infinite series S — is convergent when 



the point u lies inside the tetrahedron. 



It follows at once that the series will be convergent when the point lies 

 inside the tetrahedron formed by the points - vl, - 0u', - 8 2 u', and the origin. 



VII. If we write ; in the form 



ii - u 



1 L v! vf n 



-1 + — + ...—- + .. 

 u f u u" 



u 

 and write - =//+ q'O + r'd 2 it follows, as in par. VI, that the infinite 



series is convergent when p\ q', r' are all positive and their sum is less than 



