71 6 
MR. R, C. ROWE ON ABEL’S THEOREM. 
where f is a rational (but not necessarily integral) function of x and y, while y is 
determined as a function of x from the equation 
x(y) =?/' +Pn~jy n ~ ] +p,_o?/ _2 4- ... +p l y+p 0 =o 
the p’s being rational integral functions of x. 
This is clear since any explicit irrational function is the root of an equation with 
integral and rational coefficients, in which, by a suitable change of variable, the highest 
coefficient can be made unity. 
4. The shape in which it is most convenient to deal with f(x, y), and in which we 
shall in future assume it to be expressed, is obtained when its denominator is made the 
product of x'(y )—the differential coefficient of y(y) with respect to y —and a function 
of x only. 
This can always be done ; for let 
/(*> y)= 
F iQ, y) 
f 2 Gl y) 
f i(-l v ) x ( y ) 
f 3 0l y)x(jj) 
fivL y)x (y ) F aC'B y 2 ) F 2 Ql y$) ■ ■ • f 3(^- ; j y») 
x(y)^o(x, y x ) F s (®, y 2 )F s (x, y s ) . . . t\(x, y u ) 
y 1} y. 2 . . . y n being the n roots of the equation 
x~o 
and therefore functions of x ; and y x being the root which we have before denoted 
ky V- 
Now the product F 2 (,r, y x ) . . . Fo(.t, y n ) involving only symmetrical functions of the 
y s may be expressed as a function of x only ; while, using the equations 
and, lastly, 
the product 
ty r —ty r —y l 
r=2 r=1 
V = 71 S=/l 1 S = 11 T— 
- ty r y s =t ty r y-y l ty r 
r=2 s=2 r=l s=l r= 2 
&c. = &c. 
y yy?, ■••!/ , 
. (— y% 
Vi 
y»)F 2 (®, 2/ 3 ) • • • F 2 (x, y,) 
can be expressed as a rational function of x, y ; while F,(,c, y) and y'(y) are rational 
and integral functions. 
