254 Proceedings of the Royal Irish Academy. 



If a point is to remain unaltered, then we must have 



yi = pxi, Vi = px2, ys = p^3, l/i = p^i, 

 and accordingly, 



n-p, 



12, 



13, 





14 



21, 



22 -p, 



23, 





24 



31, 



32, 



33- 



- P, 



34 



41, 



42, 



43, 





44 -p 



This however cannot differ from 

 1 



11 - 



12, 

 13, 

 14, 



21, 

 22 - 

 23, 

 24, 



1 



31, 

 32, 

 33 - 

 34, 



41 

 42 

 43 

 44 



= 0, 



because the transformation is orthogonal. 



Hence the equation for p must be a reciprocal one, and we have 

 accordingly 



p* + 4^p3 + 6Bp~ + 4^p + 1 = 



of the tetrahedron whose vertices are given by the four values of p, 

 four of the edges from two pair of generators common to the surfaces 

 = and 17= 0. Let aj, ag, a^, Ui be the coordinates of one corner of 

 the tetrahedron corresponding to p, so that we have 



(ll-p)ai+ 12a2 + 13a3 + 14a4 =0, 



21 ai +(22-p)a2+ 23a3 + 24a4 =0, 



31 ai + 32a2 + (33 — p)oL3 + 34a4 = 0, 



41ai 



+ 42ao 



+ 43a3 



+ (44 - p) ai = 0. 



Let (3i, fSi, ySy, ^4 be the other corner of the tetrahedron character- 

 ized by the circumstance that a/3 is not a generator of Q, then we 

 have 



1 



11 - -h8i+ 12/3o + &c. = 0, 



PJ 



and similar equations, as it is obvious that the reciprocal values of p 



