318 
Proceedings of the Royal Society of Edinburgh. [Sess. 
and the anharmonic relation. The determinant “SP thus again makes its 
appearance, and associated with it is the determinant 
1 a + a act / 
1 /? + / 3' /S/S' 
1 y + y' yy 
or Y say 
for the reason that, when S = a and S' = a, is readily shown to be equal to 
(a! — a)Y. 
To obtain the required non-determinant forms of the two the multipli- 
cation-theorem is used with pleasing effect. In the first place Y is multi- 
plied row- wise by 
u 2 — u 1 
V 2 -V 1 
the result, being of course, 
Y . (iu - v)(^w - u)(v - u) — 
w 1 
( u - a)(u - a) ( V — a)(v - a) ( w - o)(w - a) 
(u - p)(u - P) ( v - p)(v - P) (w - p)(w - p) 
(u - y )(u - y) ( V - y)(v - y) (tv - y ){w - y) 
In this Cayley then puts u = a, v = a, obtaining 
Y . (a — a) = (a - /3)(a - P)( a - y)(a - y) - (a - p)( a - p)( a - y)(a - y), 
and putting u,v,w = «, /3, y obtains 
Y = ( a - P)(P - y)(y - a) - (a - y ’)(P - a)(y - p) , 
— a result known to Hesse in 1849 (see Grelles Journ., 1. p. 265). 
In the next place (§ 114) "T" is multiplied by the similar determinant 
the result being 
ss -s' - s 1 
tt' -t’ -t 1 
uu — u —u 1 
VV —V - V 1 
or 4>' say, 
(s - a)(s' — a) (t - a )(t' — a) (u — a )(u — a) ( V - a)(v — a) 
3’) {t-m’-P) <« - /S)(«' - /S') (v-p)(vmp) 
(s - y)(s’ - y) U - y)(t’ - y) (u - y )(«' - y) ( u - y)(n’ - y) 
(*-«)(*' -S') S') (u-8)(u‘- S') (w-S)(»'-S') 
so that on putting 
s,t,u,v | ) a, p,y,6 
s' ,t' ,u ,v j J P , a , S' , y 
a, P,y,8 
