284 
ON THE DERIVATIVES OF THREE BINARY QUANTICS. 
[681 
Also, attending as before only to the terms in a, and therein to the 
power of x, we have 
Q(R, Py = a 2 e 3 [g] 4 , 
R(P, QY = a 3 e 2 -r- [r] 4 ; 
that is, 
[qY Q (R, py = U, [r] 4 R (P, QY = U""; 
highest 
and, observing that [q + r — 6, 01...6} is = [_q + r] 7 , and that [q+r—Q, 456}, &c., may 
be written [q — r, 210}, &c., where the superscript bars are the signs —, the formulae 
become 
{q+r, 2l0}(l), {q+r, 510(2), {q+r, 630}(3), {q+r, 651}(4), {q+r, 654}(5), 
[q+r] 7 [qYQ(P, Ry= [qf , +4[i] 3 , + 6 [q] 2 , + 4>[qY , +1 
[g , +r] 7 [r] 4 P(P, QY= [r] 4 , - 4 [r] 3 , + 6[r] 2 , - 4 [r] 1 , +1 
Written at full length, the first of these equations (which, as being the fourth in 
a series, I mark 4th equation) is 
[g-f r] 7 [g-] 4 Q(P, P) 4 = 1 .q+r .q+r —l.q+r—%. [p] 4 [<?] 4 . P, (Q, P) 4 (4thequat.) 
+4.q+r .q+i—l.^+r—5.[^3 —l] 3 [g] 3 [g—3] 1 [r — l] 1 . (P, (Q, R) 3 ) 1 
+ 6.q+r .q+r—%.q+r—6.[p—2] 2 [g , ] 2 [g r — 2] 2 [r — 2] 2 . (P, (Q, P) 2 ) 2 
+4>.q+r— l.q+r— 5.q+r—6.[p—3] 1 [g'p[<2— l] 3 [r — l] 3 . (P, (Q, P) 1 ) 3 
+ l.q+r— l.q+r—5.q+r—6. [<?P [r] 4 . P, (Q, P) 4 , 
and the other is, in fact, the same equation with q, Q, r, R interchanged with 
r, R, q, Q; the alternate + and — signs arise evidently from the terms 
(P, QY, =(Q, Ry; (P, QY, =-(Q, Ry-, &c., 
which present themselves on the right-hand side. 
It will be observed that the identity has been derived from the comparison of 
the terms in a, which are the highest terms in x, the other terms not having been 
written down or considered; but it is easy to see that an identity of the form in 
question exists, and, this being admitted, the process is a legitimate one. 
The preceding equations of the series are 
[q + 
r] 1 [?p Q(P, P) 1 = 
1. [pp [g] 1 
P(Q, RY 
(1st 
equation) 
+ 
1 . 
[?P 
M 1 
(P, QRY; 
I\q + 
r] 3 [q] 2 Q(P,RY = 
: 1. 
, q+r 
. [p] 2 [q] 3 
P, (Q, RY 
(2nd 
equation) 
+ 2, 
. q+r- 
-l.[p- l] 1 [q] 1 [g 
— I] 1 [r 
-ip 
(P, (Q> RY) 1 
+ 1 
.q+r- 
-2. 
[qY 
[r] 2 
(P, QRY-, 
lq+ 
r] 5 [q] 3 Q(P, Ry= 
1. 
q+r 
.q+r—1. [p] 3 
[q] 3 
P, (Q, RY 
(3rd 
equation) 
+ 3. 
q+r 
.q+?—3.[p-l] 2 
[?]*[?- 
2] 1 [r—! 
2p (P, (Q, Ryy 
+ 3 
.q+r- 
-1 .q+r — 4. [p— 2] 1 
■[??[?- 
•1] 2 |>- 
i] 2 (P, QRf 
+ 1 
. q+r- 
-3.g+r—4. 
№ 
[r] 3 
(P, QRf.