286 ON THE DERIVATIVES OF THREE BINARY QUANTICS. [681 
which is easily verified, term by term; for instance, the terms with a, ¡3, or 7, give 
[q + r] 3 [q] 2 pr (p + r — 2) = (q + r) [<qj 2 [p] 2 qr (q + r — 2) 
+ (q + r - 2) [g] 2 [r] 2 p{q + r)(p + q + r- 2), 
which, omitting the factor (g'+ r) +r — 2) [^] 2 ^r, is 
(g + r - 1) (p + r - 2) = (p - 1) q + (r - 1) (p + q + r + 2); 
viz. the right-hand side is 
(p — 1) q + (r — 1) q 4- (r — 1) (p + r - 2), = (q + r — 1) (p + r — 2), 
as it should be. 
The equations are useful for the demonstration of a subsidiary theorem employed 
in Gordan’s demonstration of the finite number of the covariants of any binary form 
U. Suppose that a system of covariants (including the quantic itself) is 
P, Q, R, 
this may be the complete system of covariants; and if it is so, then, T and V 
being any functions of the form P a Q^Ry..., every derivative (T, V) d must be a term 
or sum of terms of the like form P a QPRy...-, the subsidiary theorem is that in order 
to prove that the case is so, it is sufficient to prove that every derivative (P, Q)°, 
where P and Q are any two terms of the proposed system, is a term or sum of 
terms of the form in question P a Q p Ry.... 
In fact, supposing it shown that every derivative (T, V) 6 up to a given value 
6 0 of 6 is of the form P a Q^Ry..., we can by successive application of the equation 
for Q (P, R) 6+1 , regarded as an equation for the reduction of the last term on the 
right-hand side (P, QR) 6+ \ bring first (P, QR) e+1 , and then (P, QRS) e+1 ,.., and so 
ultimately any function (P, F) 0+1 , and then again any functions (PQ, V) e+1 , 
(PQR, V) e+1 ,.., and so ultimately any function (T, V) 9+ \ into the required form 
P a Q p Ry...: or the theorem, being true for 6, will be true for 0 + 1; whence it is 
true generally.