46 PROFESSOR CAYLEY’S NINTH MEMOIR ON QITANTICS. 
may be represented by y 15 y 2 , y 3 , . . the covariants of the fourth degree written in the order 
(r,/)”, (7/), (y,/T ■ • • (r «/), (7/r ■ • (y s /)". (7/), (7./? • ■ • 
may be represented by h 3 . . ., and so on: we thus obtain in a definite order the 
covariants of a given degree m ; say, these are ^ 4 , (m 2 , g> 3 , ^ 4 , . . . . : any term is said to 
be a term than the preceding terms y> 15 ^ 2 , and an earlier term than the following 
ones, |«; 5 , &c. 
Observe that each term is a derivative the derivatives of an earlier X are 
earlier than those of a later X ; and as regards the derivatives of the same X, the deri- 
vative with a less index of derivation is earlier than that with a greater index of deri- 
vation, or, what is the same thing, those are earlier which are of the higher order. 
352. The series p 1? y> 2 , /x 3 , ^ .... is not asyzygetic ; we make it so, by considering in 
succession whether the several terms ^ 2 , (jj 3 , . . . respectively are expressible as linear 
functions of the earlier terms, and by omitting every term which is so expressible. The 
reduced series thus obtained is called T„ T 2 , T 3 , ... Observe that not every is a T, 
but that every T is a (m ; every T therefore arises from a derivation upon f and a certain 
term X; which term X (supposing the X series reduced in like manner to $,, S 2 , S 3 , . . .) is 
a linear function of certain of the S’s. Each later T is derived from later S’s, or it may 
be from the same S’s as an earlier T ; viz. if the later T is derived from (S 1? S 2 , .... S e ), 
then the earlier T is derived, it maybe, from (S n S 2 , . . . S„), or from (S 15 S 2 , ... S 0 _*), but 
so that there is not in the series any term later than S 0 . 
And if, considering any T as thus derived from certain of the S’s, and in like manner 
each of these S’s as derived from certain of the R,’s, and so on, we descend to any pre- 
ceding series, 
M n M 2 , M 3 
it will appear that the T is derived from a certain number (M,, M 2 , . . . M 0 ) of the terms 
of this series. 
353. The quadricovariants {ff)\ {ff)\ {ffYi • • • are °f different orders, and conse- 
quently asyzygetic. They form therefore a series such as the T-series, and they may be 
represented by 
B„ B 2 , B„ 
Supposing/' to be of the order w, B 4 is of the order 2 n, B 2 of the order 2n— 4, B 3 of the 
order 2n — 8, and so on. Those terms which are of an order greater than n, are said to 
be of the form W (agreeing with a subsequent more general definition of W); those which 
are of an order equal to or less than n, are said to be of the form % ; so that the earlier 
terms of the B series are W, and the later terms are %, ; viz. the ;/ terms taken in order, 
beginning with the earliest, are % 2 , % 3 , . . . 
354. By what precedes any particular T is derived from certain terms B 1} B 2 , . . . . B 0 , 
of the B series. This series, B 1? B 2 , . . . B 0 , may stop short of the terms or it may 
include a certain number of them, say % 2 , . . . The terms derived from the %’s are 
in the sequel denoted by P r 
