ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
331 
Each of the equations X=0, Y=0 has mn roots; if x—x M , y=y lx > be a pair which 
make X ari d 0 both vanish, these may be called congruous; but if x=x IJL , y=y^ do 
not make x and 6 both vanish then for these 
A = 0, 
while for congruous values A does not vanish. 
Moreover, for congruous values of x and y we at once have 
dX dY 
dx dy 
A.J(y, 0) 
the congruous values being substituted for the variables, so that 
1 _ A 
J(y, 6) dX dY 
dx dy 
Now in (v) the summation is for all the x’s and for one of the y s, say y l M , which 
may be regarded in the following way. When the equation 
X (x, y) = 0 
is solved for y in terms of x, there will be n roots ; take one of these and denote it 
by y x , which is therefore a function of x. Substitute in turn x y , x 2 ,... , x mn ; then we 
obtain for y x a series of values, but all derived from the single root of y. Thus 
„ T SO 
v__ 
X—U J(y, 6) 
~x -K»dX dY 
dx dy 
since we have x=x^ y=y hll (y=l, 2, , mn ) as the mn congruous roots. Moreover 
for roots other than these 
A = 0 
so that we may add on a number of vanishing terms to the right-hand side, and the 
removal of the restriction now gives 
(where y'=l,y or 2,y or . 
Moreover from (iv) 
and therefore 
T 
« ot >c£X dY 
dxn dy^ 
or ii,y). 
a x =dx-by 
_ n dY 
