ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
339 
and integrals connected with this curve are taken, the upper limits assigned being the 
abscissee of its points of intersection with another curve, the equation to which 
6(x, y) = 0 
contains a number of variable parameters and therefore represents a variable curve. 
But the more general form of the theorem extends the application to curves in 
space. We take the curve which is the intersection of two surfaces 
F,„(x, y, z) — 0 
Y„(x, y,z) = 0 
(and which will, as a rule, be a tortuous curve), and forming the corresponding 
integrals we assign as the upper limits of these the ordinates x of the points of inter¬ 
section of this tortuous curve with a surface the equation to which 
Yj{x, y,z)= 0 
containing a number of variable parameters represents a variable surface. 
The discussion of this geometrical interpretation and of the deductions to which it 
leads has been carried out in a memoir by Clebsch (‘ Crelle,’ t. lxiii., p. 189, 1863), 
wherein he proceeds from the theorems which are the forms of (iv) and (v) when the 
right-hand sides are zero. Example I. which follows was suggested by an analogous 
geometrical illustration which Professor Cayley gave in one of his lectures at 
Cambridge in the Michaelmas Term, 1881, wherein he pointed out how to obtain 
sn(M-pr-j-'ic) from the analytical expression for the co-planarity of the four points of 
intersection of an arbitrary plane (corresponding to E^O) with a fixed tortuous curve 
in space which was the intersection of a circular cylinder and an elliptic cylinder 
respectively corresponding to F w =0 and F„=0. 
We now proceed to consider two examples of (iv). 
11. Example I. 
Let F /tt =/—(l-ar) = 0 
F„=F-(1-£V) = 0 
Fj, = A.r-f- B y -fi Cz — 1. 
The eliminant X is obviously 
X=U{Ax-l±B(l-xJ±C{l-Wf} 
II denoting the product of the four expressions which the above includes owing to the 
two double signs. It is evidently of the fourth degree in x\ let the roots be aq, aq, 
x s> aq,. As there are three arbitrary constants there will be one relation between 
these four roots, and this can be exhibited in the form 
2x2 
I 
