1898.] of a certain determinant, etc. 5 



Hence we have generally 



Vij = cos aij and r'# = cos a\j , 



where a^ is one of the two supplementary distances between the 

 vertices i,j and a'y one of the two supplementary angles between 

 the corresponding faces. 



The choice made between the supplementary angles and 

 distances is such that when two vertices i, j coincide, and 



r t j = COS ttij= 1, 



then r'ij = cos a'y = — 1, and conversely. 



In the case of n = 3 this is equivalent to taking a, b, c and the 

 exterior angles tt — A, it — B, w — C as the sides and angles of 

 the spherical triangle. 



The Jacobian J can now be interpreted as the Jacobian of 

 the angles between the faces of a polyhedron with respect to 

 the lengths of the edges, multiplied by a simple function of the 

 sines of these angles and lengths. 



The relation between distances and angles in elliptic space 

 being known to be reciprocal it follows that r# is formed from the 

 quantities r' by the same process by which r'y is formed from the 

 quantities r. We can use R', R/, R^', &c. to denote the same 

 functions of the quantities r^ as are R, Ri, Rij of the quantities r^ 

 and in any formula we can now interchange dashed and undashed 

 letters. 



§ 3. Statement of the result and reduction of J to a quotient 

 of two simpler Jacobians. 



It is easy to verify that in the cases of n = 2, n = 3, the required 

 Jacobian is 



(_ iytn (n-i) (i^-2/ni? .} * (n+i) (2). 



1 



This result will now be shewn to be true for n if it holds for n — 1, 

 so that its general truth will follow by induction. 



Let the relations between the \n{n— 1) quantities r v - and the 

 %n(n — 1) quantities r\j be expressed by two groups of equations : 

 viz. ^ (n — 1) (n — 2) equations of the type 



% q = -r' pq +(ji(r u> ...) = 0, p, q = l, 2, ..., n-1, 



and n — 1 equations of the type 



F in =-r in -rf(r' i2> ...) = 0, i = l, 2, ..., n-1. 



