681] 
ON THE DERIVATIVES OF THREE BINARY QUANTICS. 
283 
(5) 
[p - 4]° [g] 4 [y] 4 (P, QRy = 
[■ZJ 4 
• &2®3 
(U""), 
+ 4 [2] 3 [r] 1 
. d,b 3 
(- U'"), 
+ 6 [(¿Y [r] 2 
• c 2 c 3 
(V"), 
+ 4 [2] 1 [r] 3 , 
> b 2 d 3 
(- U'), 
+ M‘ • 
CL 2 6 3 
(U). 
Thus, for the second of these equations, 
(P, (Q, R) 3 ) 1 = d x P. d y (Q, R) 3 &c.; 
the term in a x is d y (Q, R) 3 , =(d x Q, R) 3 + (Q, d y R) 3 , the whole being divided by [p— l] 3 ; 
where attending only to the highest terms in x, the two terms are respectively 
(b%d 3 3c 2 C3 "I - Sd 2 b 3 e 2 a 3 ) — (y 3]\ 
and 
(a 2 e 3 - 3b 2 d 3 + 3c 2 c 3 - d 2 b 3 ) + [5 — 3] 1 , 
which are each divided by (jo—l] 3 as above; whence, multiplying by 
[p - !] 3 [q -1] 2 [r -1] 1 , 
we have the formula in question; and so for the other cases. 
Writing now (1), (2), (3), (4), (5) for the left-hand sides of the five equations 
respectively; and 
-F', F: 
Z", Z\ Z: 
- W"', W", - W', W: 
U"\ - U"\ U”, - U\ U: 
for the literal parts on the right-hand sides of the same equations respectively; 
then we have 
X= F + F, 
Y=Z+Z', Y' = Z' + Z", 
&c., 
and the equations become 
(1)= * 
(2) = [r — l] 1 F — 1 [q-3] 1 Y' 
(3) = [r - 2] 2 £ - 2 [r - 2] 1 [q - 2] 1 Z' + l [q- 2] 2 X' 
(4) = [r - l] 3 W - 3 [r - l] 2 [q - l] 1 W' + 3 [r — l] 1 [q - l] 2 W" - 1 [q - l] 3 W'" 
(5) = [r] 4 U — 4> [r] 3 [qY U' + 6 [r] 2 [q] 3 U" - 4 [r] 1 [g] 3 U"' 4 [g] 4 U"", 
which are, in fact, the equations considered at the beginning of the present paper, 
putting therein a = r-3 and /3 = q- 3, they consequently give 
(g+r-6, 456}(1), (2+r-6, 156}(2), {q+r-6, 036}(3), [q+r-6, 015}(4), {q+r-6, 012}(5), 
6,01...6}P = [q — 3, 0123} ,+4{gr-3,123}, +6(2-3,23}, +4(5-3,3} 
„ } U""= [r — 3, 0123} , -4{r-3, 123} , + 6 (r — 3, 23} , -4(r-3,3} 
+ 1 
+ 1 
36—2