144 
Proceedings of Royal Society of Edinburgh. [sess. 
a + 
b b . 1 \ (a + Jo? + 4&Y+ 2 ( a - Jd 2 + 4 b\ n +* } 
a + a + . . .(n b's) y« 2 + 4&|\ 2 ) \ 2 / j 
= « n+1 + C^a^b + C w _ 1(2 a n ~ 3 b 2 + 
which by putting a=\ = b give the number of terms in Y n , — a 
number also obtained in the form 
+ C w+2i 3*5 + C n+2 , 5 ‘5 2 + 
1 i C 
~ 2 ^+ 1 ^ n + 2 - 1 
Anything else is of small moment. 
Cayley, A. (1857, April). 
[On the determination of the value of a certain determinant. 
Quart. Journ. of Math., ii. pp. 163-166 ; or Collected Math. 
Payers , iii. pp. 120-123.] 
The determinant in question is rather more general than Syl- 
vester’s of the year 1854 ( Nouv . Annates de Math., xiii. p. 305), 
being 
<9 
1 
... 
X 
6 
2 
. . . 
x-l 
6 
3 
... 
x - 2 
6 
... 
. 
... 6 n - 1 
# 
... x-n + 2 0 
while the other is obtained from this by putting x = n - 1. De- 
noting his own form by U„, Cayley, with Sylvester’s results before 
him, found 
U 2 = (0 2 -1) - (a-1), 
U 3 = 0(6 2 - 4) - 3(aj-2 )6 , 
U 4 = (0 2 -l)(0 2 -9) - 6(x - 3)(0 2 - 1) + 3(x-3)(x-l); 
so that, if he put H w for the value of U w in Sylvester’s case (viz., 
when x — n- 1), he could write 
V 2 = H 2 -(*-l)H 0 
U 3 = H 3 -3(a:-2)H 1 
U„ = H 4 - 6(a: - 3)H 2 + 3(a; - 3)(2; - 1)H„ , 
