ABEL’S THEOREM AND ABELTAN FUNCTIONS. 
335 
Now the denominator will consist of symmetric functions of the y s and z’s, the 
coefficients in its different terms involving x. These can he expressed in terms of x 
alone,* while the new term in the numerator can be expressed in terms of symmetric 
functions of the y s and z’s and of y x and z x , and thus T is reduced to the form 
and therefore 
■XT dw=t 
IT 
M 
u 
BF 
fix) ^p 
\ x, y, z 
Let x—cl be any root of _/(«:) = 0 ; then, as before, we consider 
(ii). 
that is, 
x—u 
vJL ^ 
X — a .1 
the summation being for the mnp values of x ; and a definite value of y and one of 2 
are to be substituted in terms of x before the summation is effected. 
Having these definite values of y and 2 (obtained from F,„=0, F„=0) if in them we 
substitute in turn the mnp values of x, we shall have mnp congruous values and 
therefore all the congruous values. For these, as we easily see, 
and therefore 
_ 1 _A 
I F m , F„, F„ \~~dZ dXdY 
\ x, y, z ) dz dx dy 
• 1 dw=% 1 SF„ ,- 7 -f t- . 
x—a x — a AxZ dX. dY 
ctedxdy 
(Hi) 
the summation on each side being the same. But for all values not included in this 
summation we have A = 0, and therefore the restrictions on the right-hand side may be 
removed without altering its value, and we shall consider the summation to extend 
over all the roots of F OT =0, F„ = 0 considered as equations in y and z and over all the 
roots x. 
Let a denote the minor of A, /3 that of B, y that of C (in each case with the same 
suffix) in A. Then we have 
04 >»X fi- /3 m \ -}- y m Yj =AF ffl 
«/(X+/3«Y -by„Z = AF„ 
o^Xffi/^Y -\-y p r L. = AF ? ,. 
* Cf . Salmon’s ‘ Higlier Algebra,,’ § “4. 
