330 
PROFESSOR A. R. FORSYTH ON 
or writing 1 
O 
the equation (ii) becomes 
or 
and therefore 
^dx+^dy=0 
dw= 
b JC 
b U 
dy 
b X 
<b. v 
— dw 
/Ay b6 bd t>y' 
\bx by b.v by 
+ Sd=0 
dio= 
bd _ 
• T (y> 
t\Jdw=t 
U 
J (x> ^ 
Sd 
(iii) 
where U is any rational function of x and y, and the summation is taken over all the 
roots ay of the equation obtained by the elimination of y between y and 6. 
Let X, Y respectively denote the eliminants of y, 6 with regard to y, x; then we 
can express X, Y in the form 
X=A X +B0 1 
> .(iv) 
Y=Cy + DdJ 
and we write 
A = AD-BC. 
Now whatever the function U may be it can be written in the form 
f\x) ’ 
for it must be expressible as 
fdx, yd 
/i(A yd 
that is, 
l u.=n 
M x > yd n A( x > Vv-> 
_ v-= 2 _ 
’n/iO, yd) 
^=i 
which by means of the equation y—0 is at once reduced to the above form; thus 
.iv). 
_ T <fc „ T Sff 
V--X-_ 
A x )bc /(«)•-%> 0) 
by 
