ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
333 
„ t dx ^ t 
V---;N 
80 
\f{x) *>x y(x) j( x , 0) 
*>y 
= 0 
L M. 
T 80 
* h X Q 
by 
where the summation on the left-hand side is over the roots of the equation 
X = 0 
while on the right-hand side it is over the n roots y of the equation x (x, y) = 0. The 
variables on the right-hand side being the arbitrary constants in 6 which occur only in 
$0 
the factor -y, we may integrate, and we have as the result 
fV T dx 
" 1 - 
_/(U_ 
by 
y=y * f T 1 
S z~ log 0 +0 
y=yi ! S | 
I'ty J 
agreeing with the form given in Professor Lowe’s memoir (Phil. Trans. 1881, p. 721). 
7. In the generalisation of the theorem we shall consider only two dependent 
variables y, z and one independent variable x ; it will be seen that the work would 
apply, mutatis mutandis, to k—\ dependent variables and one independent. Let the 
variables y and 2 be given as functions of x by the equations 
F m (x , y,z)=0 
(x, y, z) = 0 
of the degrees m and n respectively. Let 
y (x, y, z) 
be a function of x, y and z the coefficients of y and z in which are functions of x with 
any number of arbitrary constants ; so that as in the simple case when z, x and y 
vary the constants also vary. Using the same notation as before we have 
Therefore 
^ dx +T7j dy+ 
'-—d .'-+~ du-\- 
bx by J 
bF m 
bz 
bF u 
bz 
dz— 0, 
dz— 0 . 
dz dx dy 
b(F„„ F a ) t(F w , F w ) t(F w , Fj 
b(x, y) b{y, z) b(z, x) 
and 
