ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
327 
the \p(p —1)+ • • • equations; then the eliminant is the quotient of the determinant 
formed by any 
l(m+ n +p— 1) (m +n +p) 
rows of the set first written down by the determinant formed from the second set 
after the elision of these rows. 
3. To show how this can be brought into the desired form the easiest plan will be 
to consider an example. Let 
F^Ay+Bs+C 
F 2 =Ay+ Yyz +B V+E ,'y+ Vz+C' 
F 3 =AY+D , V 3 z +E> 3 +B /, f 5 +F / y+G>+JV+H"y+K ,, z+C" 
where the coefficient of the highest powers of 2 and y are constants and those of 
other powers are functions of x such as make the order of the highest expression in 
the term of the same order as the equation ; thus, for instance 
F "=/&+/' 
H"= hx 2j r h'x + h" 
and so on. Then we have, since m— 1, n—2 , p— 3 
2/ 4 
yh 
2/z 2 
yz* 
^4 
yz 
F 3 
z 3 
>r 
\) z 
z 2 
y 
2 
1 
2/ f 2 - f i 
zFo.Fj 
f 2 .f x 
F 3 F 1 
r F i 
A 
B 
c 
A' 
A" 
f-zF, 
A 
B 
C 
F' 
A’ 
D" 
yz% 
A 
B 
C 
B 
F' 
E" 
A 
B 
C 
B’ 
B" 
A 
B 
c 
K 
A' 
F" 
2/ zF i 
A 
B 
C 
D' 
E’ 
F' 
G" 
z% 
A 
B 
c 
D r 
B' 
J" 
y$ i 
A 
B 
c 
C 
E' 
H" 
A 
B 
C 
C’ 
D' 
K" 
F i 
A 
B 
C 
C 
C" 
2/ 2 F 3 
A’ 
F 
B' 
E' 
D' 
C' 
A 
2/ zF 2 
A' 
F' 
B' 
E' 
D' 
C' 
B 
A 
^f 2 
A' 
F' 
B' 
E' 
D' 
C' 
B 
2/F 3 
A' 
F' 
B' 
E' 
D' 
C' 
C 
A 
zF 2 
A' 
F' 
B' 
E' 
D' 
C' 
C 
B 
F 2 
A' 
F' 
B' 
E' 
D’ 
C' 
C 
2/ F 3 
A" 
D" 
E" 
B'' 
F" 
G” 
J" 
H" 
K" 
C" 
A 
zF 3 
A" 
D" 
E" 
B" 
F” 
G" 
J" 
H" 
K" 
C" 
B 
F 3 
A" 
D" 
E" 
B" 
F" 
G" 
J'' 
H" 
K" 
0" 
C 
To find the eliminant we choose any 15 rows (leaving out say the y 2 F x , yzF v zF 2 , 
2/F 3 ) and form a determinant, and then divide by 
