1920-21.] A Continuant of Cayley’s of the Year 1874. 
Ill 
XI. — Note on a Continuant of Cayleys of the Year 1874. 
By Sir Thomas Muir, F.R.S. 
(MS. received March 8, 1921. Read March 21, 1921.) 
(1) In a ‘‘Note sur une formule d’integration indefinie” of the year 1874,* 
Cayley has occasion to use a determinant of a very peculiar structure, 
whose value when of the (^ + l) th order is 
{[a]# 2 * + [&]z/ 2 } n , 
[a] r in the development of this standing for 
a(a— 1) . . . . ( a — r+ 1). 
The first three instances are 
1 (a— \)x— by 
2y (a - l)(x 2 Pxy) 
V 2 
1 
y 
ax — by 
a(x 2 + xy) 
= ax 2 + by 2 , 
1 
(a + \ )x— (b - 1 )y 
a(x 2 + xy) 
= a(a — l)# 4 + 2 abx 2 y 2 + b(b— l)y\ 
1 ( a—2)x—by 1 
3 y (a—2)(x 2 + xy) ax-(b—\)y 2 
3 y 2 . (a - \){x 2 + xy) (a + 2)x - (b - 2)y 
I y B . . a(x 2 + xy) 
= a(a-l)(a-2)x G + 3a(a-l)bxY + 3ab(b-l)x 2 y 4 + b(b - l)(b - 2)y\ 
where, it is as well to note, the coefficients of x 2 -\ -xy in the main diagonal 
increase by 1 at each step, and the coefficients of x in the adjacent 
minor diagonal increase by 2. Viewed as an equivalent for a power of 
\_a\x 1 + [b~]y 2 the form of the determinant is far from attractive, its order 
being higher than seems natural, and there being nothing in it to show 
that it is unaltered by the change of x into — x, of y into — y, or by the 
simultaneous interchange of a with b and x with y. 
(2) Resembling it, but more pleasing in form, is another determinant, 
also noted by Cayley the first three instances of which are 
* Comptes rendus . . . Acad, des Sci. (Paris), lxxviii, pp. 1624-1629 : or Coll. Math . 
Papers, ix, pp. 500-503. 
t The reference for this I cannot at present find. 
