332 
PROFESSOR A. R. FORSYTH ON 
the term corresponding to a differential of X disappearing and the others vanishing m 
virtue of the values assigned to the two variables; thus our expression becomes 
— 5 1 
T„ 
se„ 
Xu . — a 
6% dX 
by^r dx„ 
1 y . 
But the coefficient of — in the expansion of — in descending powers of x is 
OC -A. 
Cit 
and therefore the foregoing 
O O 
dX 
(*-*>*; 
_ s v 
* dX 
dx, 
T Td 
= —Ci 2 —— 8(9 
«—« 
by 
"V ^ 
the S referring to the n values of y obtained from the equation (i), and the expansion 
being in the factors of X alone. But since we are substituting for y from (i) we have 
X always zero in this, and therefore 
X=B^. 
Taking now into account the expansion for the factor > we have finally 
X — a 
80 
J (x e ) 
-Cit 
x 
T 
X — a 
*°± + t 
* ^ 
X &ft 
bx 0 
J>y 
the summation in each of the terms on the right-hand side being for the n values of y. 
Now on the introduction of Boole’s symbol 0 (cf. Phil. Trans., 1857, p. 751), the 
right-hand side is merely the definition of 
_x —« 
T 80' 
bx 0 
8y 
Let x—rx be a root of fix) = 0, and in (v) expand [/(#)] 1 in a series of partial 
T 80 
fractions corresponding to the roots ; then expressions of the form -— J(f~0) are 
obtained. Moreover, from the nature of the preceding fractions and the definition 
of the symbol © in connexion with them, it is obviously a distributive symbol; thus 
we have 
