MR. A. CAYLEY’S INTRODUCTORY MEMOIR UPON QUANT1CS. 
253 
the operation 2(^1— xd 9 will be equivalent to {xb s }-\-tjd s — x^ y , and therefore every 
covariant of the system must be reduced to zero by each of the operations 
23= {a?d,} xb r 
This being the case, we have 
B.O=23 ( I ) +23( < f ) ) 
0.©=<I>B+<&(23), 
equations which it is obvious may be replaced by 
and consequently (in virtue of the theorem) by 
23.<!>=23 < I > +»7df( < I > ) 
and we have therefore 23. 0— -0.28= — ({^} — ^)(0) ; 
or, since <b is a covariant of 0, we have 28. <£=0.28. And since every covariant of 
the system is reduced to zero by the operation 23, and therefore by the operation 
0.28, such covariant will also be reduced to zero by the operation 23.0, or what is 
the same thing, the covariant operated on by O, is reduced to zero by the operation © 
and is therefore a covariant, i. e. O operating upon a covariant gives a covariant. 
16. In the case of a quantic such as U= 
m m' 
(*X x > yX x ',y')~), 
we may instead of the new sets (|, r), (§', j/).. employ the sets {y, —x), {y', — x 1 ), &c. 
The operative quantic 0 is in this case defined by the equation 0U=O, and if 0 be, 
as before, any covariant of 0, then O operating upon a covariant of U will give a 
covariant of U. The proof is nearly the same as in the preceding case ; we have 
instead of the equation = the analogous equation 
^({^}) = “ {*£,}($), 
where on the left-hand side {.rd^} refers to U, but on the right-hand side {^} refers 
to 0, and instead of 28 = {.rd 3 ,}-}-jjB f — xd y we have simply 23 = {ffdj,} — xd y , 
17. I pass next to the quantic 
(*X x > y) m > 
which I shall in general consider under the form 
(a, b..b\ a'Xx,y) m , 
but sometimes under the form 
(a, b...b\ d%x, y) m , 
the former notation denoting, it will be remembered, 
ox m -{-jbx m ~'y.. +jb'xy m ~ 1 + dy m , 
