278 
[681 
681. 
ON THE DERIVATIVES OF THREE BINARY QUANTICS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), 
pp. 157—168.] 
For a reason which will appear, instead of the ordinary factorial notation, I 
write {a012} to denote the factorial a. a +1 . i + 2, and so in other cases; and I 
consider the series of equations 
(1) = X 
(2) = ({ Q 0}, i/90}$F, -Y'), 
(3) = ({aOl}, 2 {al} {/31}, {/901}$7, -Z\ Z"), 
(4) = ({a012}, 3 {al2} {/92], 3 {a2} {/912}, {/9012}$17, - W', -17", - 17"'), 
&c. 
where 
X = 7 + 7', 
Y=Z +Z', 7' = Z' +Z", 
Z = W + W, Z' = W' + 17", Z" = 17" + W'", 
&c. 
We have thus a series of linear equations serving to determine X ; 7, Y'; Z, X, 7"; 
17, IF 7 , 17", 17'"; &c. We require in particular the values of X; 7, 7'; 7, 7"; 
17, IF"'; &c., and I write down the results as follow: 
X = (1), 
(1) (2) 
{a+ /90} 7 ={/90}, +1, 
{ „ } 7' = {a0}, — 1;