Bf 
282 ON THE DERIVATIVES OF THREE BINARY QUANTICS. [681 
I recall that we have (P, Q)° = PQ, so that the first and the last terms of the 
series might have been written (P, (Q, R) 4 )° and (P, (Q, R) 0 ) 4 respectively; and, 
further, that (P, Q) 1 denotes d x P ,d y Q — d y P .d x Q; (P, Q) 2 denotes 
d x 2 P . djQ — 2d x d y P. d x d y Q + d y 2 P . d x 2 Q; 
and so on. 
I write (a, b, c, d, e) for the fourth derived functions of any quantic U, = (*$#, y) m \ 
we have, in a notation which will be at once understood, 
TJ = (a, b, c, d, e§x, y) 4 -4- [m] 4 , 
(d x , d y ) TJ = (a, b, c, d), (b, c, d, e) (x, y) 3 + [m - l] 3 , 
(d x , dy) 2 U = (a, b, c), (6, c, d), (c, d, e)(x, y) 2 -~ [m - 2] 2 , 
(d x , dy) 3 TJ = (a, b), (b, c), (c, d), (d, e) (®, 2/) 1 -4- [m - 3] 1 , 
(d*, dy) 4 TJ — (a, b, c, d, e); 
and then, taking 
(di, b 1} Ci, d x , ex), (a 2 , b 2 , c 2 , d 2 , 6 2 ), (n 3 , 6 3 , c 3 , d 3 , e 3 ), 
to belong to P, Q, R, respectively, we must, instead of m, write p, q, r for the 
three functions respectively. 
If we attend only to the highest terms in x, we have 
TJ = ax 4 -r- [m] 4 , 
(d x , d y ) TJ = (a, b) cc? -r [on — l] 3 , 
(d x , dy) 2 TJ = (a, b, c) a? -4- [m — 2] 2 , 
(d^, dy) 3 P = (a, &, c, d) x + [m — 3] 1 , 
(da;, dy) 4 P = (a, 6, c, d, e). 
Consider now P (Q, R) 4 , (P, (Q, P) 3 ) 1 , &c.; in each case attending only to the 
term in Ox, and in this term to the highest term in x, we have 
d 2 c 3 46 2 d 3 -j- 6c 2 c 3 4d 2 6 3 4- c 2 a 3 (X), 
\_q — 3p • b 2 d 3 3c 2 c 3 4* 3d 2 6 3 6 2 u 3 ( 3^), 
4- [?* 3] 1 . cl 2 q 3 36 2 d 3 3c 2 c 3 d 2 b 3 (3^), 
[<7 — 2] 2 . c 2 c 3 2d 2 b 3 4- 6 2 cl 3 (Z ), 
+ 2 [gr - 2] 1 [r - 2] 1 .6 2 d 3 - 2c 2 c 3 4- d 2 6 3 (- Z'), 
4- [V 2]-. ci 2 6 3 26 2 d 3 + c 2 c 3 (Z), 
[q-1? .d 2 b 3 — e 2 a 3 (-W"'), 
4-3[g— l] 2 [r — l] 1 . c 2 c 3 — d 2 b 3 ( W"), 
4-3[9- l] 1 [r - l] 2 . b 2 d 3 - c 2 c 3 (- W'), 
+ • [r -1] 3 . a 2 e 3 - 6 2 d 3 (IT), 
a) wmp) 4 
(2) [p - l] 3 \q - 3] 1 [r - 3] 1 (P, (Q, R) 3 ) 1 = 
(3) [p - 2] 2 [q - 2] 2 [r - 2] 2 (P, (Q, R) 2 ) 2 = 
(4) [p - 3f [q - l] 3 [r - l] 3 (P, (Q, Ry) 3 =