90 
Proceedings of Royal Society of Edinburgh. [sess. 
when the quadrilaterals are “ polar.” Let a he the pole of AB, b of 
BC, &c., then 
cos A b cos Be cos C d cos ~Da - cos Ac cos B d cos C a cos D& = 0. 
And numerous other relations can he obtained, with equal ease, 
by the same simple process. 
Cayley’s form of the expression connecting the distances, two 
and two, among five points in space is an immediate consequence 
of the identity 
tx{a - Of = %xa? - 2S02aa + &%x , 
where a l5 a 2 , Jcc., are n given vectors, 0 any vector whatever, and 
x v x 2) &c., n undetermined scalars. 
For, provided that n is greater than 4, we may always assume 
%X = 0 , %Xa = 0 , 
which are equivalent to four homogeneous linear relations among 
the a?s. 
Let, then, n — 5, and write the above identity separately for each 
a, put in place of 0. Thus we have 
%x{a — cq ) 2 = %Xo?, 
%x(a - a 2 ) 2 = ^a 2 , 
^(a - a 5 ) 2 = %Xa 2 . 
Take, with these, %x = 0 , 
and we obtain six linear equations from which to eliminate the five 
values of x. The result is, at once, A, B, C, D, E being the points, 
AA 2 BA 2 CA 2 DA 2 EA 2 1 
AB 2 BB 2 CB 2 DB 2 EB 2 1 
^a 2 = 0. 
AE 2 BE 2 CE 2 DE 2 EE 2 1 
1 11110 
As 3#a 2 may have any value, this is Cayley’s expression. An 
interesting variation of it is supplied by taking 2( xa ) = 0, instead 
of %{x) — 0, as the sixth equation. 
