228
AN INTRODUCTORY MEMOIR UPON QUANTICS.
[139
The quantic © gives rise to the symbols {£9,}, &c. analogous to the symbols {xd y }, &c.
formed from the quantic U. Suppose now that d> is any quantic containing as well
the coefficients as all or any of the sets of ©. Then {xd y } being a linear function of
a, 6, c, ... the variables to which the differentiations in <f> relate, we have
d>. [xd y \ = d> {xdy} + d> ({a#*});
again, being a linear function of the differentiations with respect to the variables
9«, 9ft, 9 C , ... in we have
№} . d> + {1/9*} (<£>);
these equations serve to show the meaning of the notations and {r?9f} (d>), and
there exists between these symbols the singular equation
*(Hr})=fo9f} m
14. The general demonstration of this equation presents no real difficulty, but to
avoid the necessity of fixing upon a notation to distinguish the coefficients of the
different terms and for the sake of simplicity, I shall merely exhibit by an example
the principle of such general demonstration. Consider the quantic
U = ax? + 3bx 2 y + 3 cy 2 + dy 3 ,
this gives 0 = g 3 d a + ^yd b + £y 2 d c + r/ s d d ;
or if, for greater clearness, d a , d b , d c , d d are represented by a, /3, y, 8, then
© = a| s + /3^ 2 q + 7 %rf + Srf,
and we have {xd y } = 369« + 2cd b + dd c ,
and {r?9|} = Sadp + 2/3d y 4- 79 5 .
Now considering $ as a function of 9«, d b , 9 C , d d , or, what is the same thing, of
a, /3, 7, S, we may write
<f> ({xd y }) = <I> (36a + 2c¡3 + dy);
and if in the expression of <f> we write a + 9«, /3 4-9&, 7 + 9„, S + d d for a, /3, y, 8 (where
only the symbols 9«, 9*, 9 C , d d are to be considered as affecting a, b, c, d as contained
in the operand 36a 4- 2c/3 4- dy), and reject the first term (or term independent of
99ft, 9 C , d d in the expansion) we have the required value of ({¿cS^)}. This value is
(9 a d> 9« 4- 9^ d b 4- 9 y 9 C ) (36a + 2c/3 4- dy);
performing the differentiations 9«, 9&, d c , d d , the value is
(3a9j3 4- 2/39 v + yds) c l ) ,
i.e. we have {{xd y }) — |r;9f} (d>).
