ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
329 
X — pyXy + pyX'y 
= A ; „F m +A«F ;{ +A y/ F \ p 
of the same form as before. 
This method of expressing an eliminant obviously admits of generalisation to the 
case of r equations in r variables. 
5. The preceding method enables us to obtain the eliminants of the equations with 
regard to the different variables in a particular form which is useful in the proof of the 
general theorem in § 7 ; but when the object is merely to obtain the equation giving 
the roots x tJ which are to form the upper limits of our integrals we should arrive at 
the result more easily as follows. 
Obviously 
[x=mn 
X= n ¥ p (x, zj 
where ?/ M , z fJ constitute one of the mn pairs of roots of the equations 
F„=0 F„=0 
regarded as giving y, z in terms of x ; and the product is taken over all these pairs. 
Now the coefficients on the right-hand side will be symmetric functions of y and 2 , and 
these can be evaluated (by the method given in Salmon’s £ Higher Algebra,’ § 74) in 
terms of x ; and there will be obtained the required equation in x. 
Abel’s Theorem. 
6. Let 
X {x,y) = 0 .(i) 
be an equation of the degree n which gives y in terms of x ; and let 
6(x, y) 
denote a function of x of degree m —reducible to degree n— 1 at most in y by means 
of (i)—the coefficients of y in which are functions of x and contain any number of 
arbitrary constants. Treating x=0, 9—0 as two equations to determine the values of 
the variables, these arbitrary constants will enter into the expressions for the values 
of x, and will therefore vary when the latter vary. Let such a variation take place, 
so that 
^c dx +^dy+ .(h) 
S operating only on the constants in 9. Moreover we have from (i) 
MDCCCLXXXIII. 2 u 
