718 
MR. R. C. ROWE ON ABEL’S THEOREM. 
x 1} x 2 , ... Xp; let the corresponding values of y, the root of ^ with which we are 
concerned, be y n , y 12 , . . . y llt * 
7. Having expressed f(x, y) in a convenient shape we have next to transform the 
dx of our integrals into the differentials of the new variables. 
If 8 denotes the operation of differentiating with regard to our new variables we 
have from the equation F = 0 by which x is connected with them 
But 
therefore 
Again 
therefore 
Y\x)dx-\- SF (x) = 0 
SE=F 0 (x)SF(x) 
SE 
dx= 
F o(®) F/ (®) 
E=%i)% 2 ) • • • %n) 
* As an example of these processes let 
X= — 1 — 
x/1 + .M 
A natural assumption is 
x(y)=!r—(. l +* 4 )=0 
so that 
/o> 
Take for the second function the form 
e (y)=y— (i+ a i x + v 2 ) 
and on elimination we find 
E(.r, a v a7)=(a^— l)rc 3 + 2a 1 a 2 ^ 2 + (a 1 2 + 2a 2 ).T + 2a 1 =0 
Now, if we had 
a 1 + a 2 = — 1 + \/2 (a linear relation) 
E = 0 would he satisfied on making x= 1 and we should have 
while 
F o(®)=®— 1 
F (,v) = (a 2 ~— 1)®- + (2a 1 a. 2 + — 1),t + cf + 2a 1 u 2 + 2 « 2 2 + u 2 — 1 
and may be expressed in terms of a l alone. 
We should also have 
/(n V) y 2)j x'(yl 
so that 
fx (*» y) =2, /°(®)=h 
