4 Mr Berry, On the evaluation [Oct. 31, 



§ 2. Connection with Elliptic Geometry ; and reciprocity between 



r and r' . 



With the methods and notation of Whitehead's Universal 

 Algebra (Bk. vl, Chaps. I., il), the reference elements e x ...e n 

 denoting the vertices of a polyhedron in elliptic space of n — 1 

 dimensions, and any point being denoted by 2^, let the equation 

 of the absolute in point-coordinates be 



& + ... + |* 2 + 2r 12 &&+...+ 2r n _ lt „ ^_^ n = 0. 

 The edge 12 meets this in the points 



or |i = - e ±tol2 ff 2 , 



where r 12 = cos a 12 ; 



.'. one of the distances between e x and e 2 is one of the values of 



|-. log (e m ™l e~ ia u) = 7a 12 . 



If therefore we take the space constant y to be unity, r 12 is the 

 cosine of one of the distances between e 1 and e 2 - 



If now we denote the faces opposite e 1} e. 2 , . . . by E 1} E 2) . . . , where 

 E x is the supplement of e x , and represent any plane by XgiEi, then 

 we know that the equation of the absolute in plane coordinates is 



i^ 2 + • • • + RJtf + 2Ri£& + • • • + ZRn-i, n %n-£n = 0, 



the quantities Ri and R^ being the minors as explained in § 1. 



Hence the planes through the edge E l E 2 which touch the 

 absolute are 



Ritf + R& 2 + 2i2 12 £ 1 f 3 = J 



y Rl2 ± V _R 12 2 — RJl 2 



or f i = ^ & 



= - e± ia '» &, 



where . 12 = cos a\ 2 = r\ 2 . 



^RA 



And hence as before one of the angles between the faces j 

 E 1 E 9 , is 



2". log (e fa/ «/*- to '«) = a\ 2 . 



