681] 
ON THE DERIVATIVES OF THREE BINARY QUANTICS. 
285 
From these four equations the law is evident, except as to the numbers subtracted 
from q + r. These are obtained, as explained above, in regard to the numbers added 
to a + fi in the { } symbols; transforming the diagrams so as to be directly applicable 
to the case now in question, they become 
0 
0 
01 
01 
012 
012 
0123 
0123 
1 
1 
2 
03 
3 
015 
4 
0127 
1 
2 
11 
14 
21 
036 
31 
0158 
2 
34 
12 
156 
22 
0378 
3 
456 
13 
1678 
4 
5678, 
showing how the numbers are obtained for the equations 2, 3, 4, 5 respectively. 
The first equation is 
(q 2 + qr) Q (P, R)=pqP (Q, R) + qr [Q (P, R) + R (P, Q)], 
viz. this is 
0 =pq P (Q, R) — qr Q (RP) +qrR (P, Q) 
+ (q 2 + qr) Q (R, P); 
or, dividing by q, this is 
0 =pP (Q, R) + qQ (R, P) + rR (P, Q), 
which is a well-known identity. 
We may verify any of the equations, though the process is rather laborious, for 
the particular values 
P = ajhiP+o-) yUp-a), Q = ¿ci(2+P) yHq-P), H — ¿cHr+y) yi(.r-y); 
thus, taking the second equation, we have, omitting common factors, 
(Q, R) 2 = q + /3 .q + /3 — 2.r — y.r — 7 — 2 
— 2 .q + (3 .q — fi.r + y.r — y 
+ .q-fi.q-fi-2.r + y.r + y-2 
= /3 2 (r 2 — r) + y 2 (q 2 — q) — 2/3y (q — 1) (r — 1) — qr (q + r — 2), 
(P, (Q, P) 1 ) 1 = (q + (3. r - y. -. q - /3. r + y) (p + a. q + r - /3 - y - 2. - .p - a. q+ r + /3+y-2) 
= (fir - q<y) (a. q + r — 2 . - p. fi + 7) 
= afir (r + q- 2) — ay q (q+r-2) —prfi- +p(q — r) ¡3y +pqy 2 , 
and from the first of these the expressions of Q (P, P) 2 and (P, QR) 2 are at once 
obtained. The identity to be verified then becomes 
[(7 + r] 3 [g] 2 {a 2 (r 2 — r) + 7 2 (p 2 — p) — 2ay (p — 1) (r — 1) — pr (p + 1— 2)} 
= (q + r) [q] 2 [p] 2 {¡3 2 (r 2 — r) + y 2 (q 2 — q) - 2fiy (q — 1) (r — 1) — qr (q + r— 2)} 
+ 2 (q + r — 1) [q\ 2 (p - 1) (r — 1) [afir (q + r- 2) - ayq (q+r — 2) 
— prfi 2 + p (q — r) fiy + pqy 2 ) 
+ (q + r — 2) [q] 2 [>] 2 {a 2 (q + r) (q+ r — 1) + (fi + y) 2 (p 2 —p) 
- 2a(fi + y) (p - l) (q + r - 1) - p (q + r) (p + q + r - 2)},