PROFESSOR CAYLEY’S NINTH MEMOIR ON QE ANTICS. 
49 
In forming the second derivative of i with the several covariants U, we may omit as 
before the four invariants, and also omit the four linear covariants a, ia, y, iy ; we have 
therefore to consider the second derivatives of i with 
/ <P, %h Y (/?)» (fp\ U' T l U 7 l (ft), (<P*)> 0')> (A ( Ti ) 
respectively : the first six of these are each of them U ; the remaining nine are each of 
the form ((PQ), if. Now ((PQ), if is a linear function of ((P if, Q), ((Q if, P), P(Q?') 3 , 
and Q(Pi) 3 . The first two of these are terms of the same form ; (Pi) 2 , as a covariant of 
a lower degree than ((PQ),*) 2 , is E(U), and hence ((Pi) 2 , Q) will be F(U) if only (TJ, Q) 
is F(U); Q being here any one of the functions/, <p, i, j, p, r, and U being any one of 
the functions 
/ <p, i,j,p, r, «, y, {ff), (, fp ), (/r), (jr), {fif {(pi) ( ji ) {pi) { n ) {ia) {iy). 
363. For U equal to any one of the last eleven values, the form is (Q, PS), which is 
=P(QS)+S(QP), and is thus a function of covariants of a lower degree; there remains 
only the derivatives formed with two of the functions/, (p, i,j,p , r, or of one of these 
with a or y. But these are all U other than the derivatives 
(fjf (<Pj)i (<pr), (P 7 )’ (f a )’ (P a f (»> CH; (fy) (<P?) (jy), (py), ( 7 y), 
and since y={ra), the derivatives containing y will depend upon covariants of a lower 
degree; there remain therefore only {fjf (<pj), {<pp), (pr), (pr) ; {fa), (<pa), [ja), {pa): 
each of these can be actually calculated in the form F(U). 
Hence finally, assuming that every covariant of a degree inferior to m is F(U), it follows 
that every covariant of the degree m is F(U) ; whence every covariant whatever is F(U), 
viz. it is a rational and integral function of the 23 covariants U. 
364. It will be observed that, writing A, B, C for P, Q, i, the proof depends on the 
theorems 
((AB), C), a linear function of A(BC) 2 , B(CA) 2 , C(AB) 2 , 
(AB, C) 2 „ ,, do. do. do. 
((AB), C) 2 „ „ ((AC) 2 , B), ((BC) 2 , A), B(AC)», C(AB)», 
which are theorems relating to any three functions A, B, C whatever. 
365. I remark upon the proof that the really fundamental theorem seems to be that 
which I have called theorem A. As to the forms W it is difficult to see a priori why 
such forms are to be considered, or what the essential property involved in their definition 
is; and in fact in a more recent paper, “Die Simultanen Systeme binaren For men ” 
(Clebsch and Neumann, t. 2 (1869), see p. 256), Professor Goedan has modified the defi- 
nition of the forms TV by omitting the condition that the order of the function shall 
exceed n ; if it were possible further to omit the condition of at least one index being 
= or<-|«, and so only retain the conditions n — i, n—j, See., each of them >0, then the 
essential property of the forms W would be that any such form was a rational and inte- 
gral function of the special covariants formed, as above, by means of the quantic of the 
next inferior order. And moreover, as regards the theorem B, there seems something 
JIDCCCLXXI. H 
