112 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
j ( a - \)x + by 1 
\ (a - l)(x 2 - xy) (a + l)x + (b — l)y 
(cl — 2)x + by 1 
(a-2)(x 2 -xy) ax + (b-l)y 2 
( a-l)(x 2 -xy ) (a + 2)x + (b - 2)y 
{[«>+[%} 2 
{[a]x + [b]yY 
(i a—3)x + by 1 
(a-3)(x?-xy) (a — \)x + (b— \)y 2 
. (a-2)(x 2 -xy) (a + l)x + (b — 2)y 
(a — l)(x 2 — xy) 
= {[a> + [%} 4 . 
(a + 3 )x + (b — 3)y 
Here the determinant is a pure continuant, and its order seems natural, 
so that the only one of our critical queries remaining is that in regard 
to the interchange of a , x with b , y. 
(3) Writing x 2 for x and y 2 for y in the second series of deter- 
minants we obtain equivalents for the determinants of the first series, 
for example, 
1 
(i a - l)a? — by 
1 
2 y 
(a- 1 )(x 2 + xy) 
(cl + \)x — (b — 1)?/ 
= 
y 2 
a(x 2 + xy) 
(a- \)x 2 + by 2 1 
(a - l)(a? 4 - x 2 y 2 ) (a + l)x 2 + (b - l)y 2 } 
so that we have two different determinant expressions for 
{[< <x\x 2 + [b]y 2 } n , 
and a not very simple-looking problem facing us in transformation. 
(4) Further, the member of the second series with the sign of y 
changed is the cofactor of the element y n in the (?i+l) th member of the 
first series ; and, following this up, we obtain the curious expansion- 
theorem 
(-1 ) n {[a]x 2 + [b\y 2 } n = y n {[a\x-[b]y} n 
- (w) 1 y n_1 {[a]aJ - \p\y } n ~ l . a(x 2 + xy) 
+ 0) 2 ; y w_2 {[a> - [%} w “ 2 • [a\ 2 (x 2 + xy) 2 
- (n)^y n ~^{[a\x - [&]y}”~ 3 . [af(x 2 + xy) 3 
(5) Before referring further to these forms of Cayley’s it is desired 
to draw attention to a new form, which, besides being interesting in itself, 
adds considerably to the interest of the others. It arises as follows: — 
Putting 
j3 1 for ax + by and 7r l for \a\x + \b]y i 
/3 2 for ax 2 + by 2 2!tt 2 for { [ci\x + [&]y}' 2 , 
jS 8 for ax^ + by 3 3!tt 3 for {[cl]x + [5]?/} 3 , 
