PROFESSOR W. K. CLIFFORD ON THE CLASSIFICATION OF LOCI. 
G79 
U 1 =w 1 +m 1 iri+g' 1 a 11 + g' 3 a 13 + . . . +q J3 a lp , 
XJo = Wg-)-7J'lo77Z -j- ^ 2 C04-7oa 2 2 “l - • • • + 
V p = u p + myna + q x a pl + q. 2 a n -f . . . +^/V 
wliere the numbers m, q are integers ; namely, m?, is the additional number of times 
the new path has gone round the hole h, and q h is the additional number of times it 
has gone through that hole. We shall write these equations thus 
U l5 U 2 , . . . U, ; — u x , u. 2 . . . u p (mod. 7 n, a), 
and shall say that the quantities U are congruent to the quantities u in respect of the 
periods 7 ri, ct. 
Suppose now that the curve has 110 actual nodes, and that a locus of any order 
intersects it in the points x x , x 2 , . . . x m . Then, if another locus of the same order 
intersects it in the points y x , y 2 , . . . y my and we take any one of the integrals, say u, 
from x x to y x , from x 2 to y. 2 , . . . from x m to y m , the sum of these results will be con- 
gruent to zero. That is to say 
xf du k = 0. 
Here the X refers to the suffixes of the x and y, not to h. There are p such equations. 
This is Abel’s Theorem. 
When the curve is in a /c-fiat and of the order n, we shall use this theorem chiefly 
for its n intersections with a (Jc — l)flat. If we regard the lower limits of the 
integrals (the points x) as fixed, the integrals for any point y may be regarded as 
parameters belonging to that point, and then Abel’s Theorem gives us p equations 
between the parameters of n points which lie on a (I: — l)flat. The truth of these 
equations is necessary to the points lying on a (/' — l)flat, but it may not be sufficient. 
Thus in a bicircular quartic curve, p=l, we have one equation to express that four 
points are in a straight line, and if the points are coll inear the equation is true. 
But it does not follow from the equation that the points are collinear ; in fact, the 
equation holds equally good if the points are in a circle. 
If the sums of the parameters of p points are given, that is, if we have the 
p equations — 
([’-}-( + . . . +| “\du h =v h li— 1, 2 , p~\, 
the v h being arbitrary constants, and the lower limits of the integrals being supposed 
constant ; then the upper limits x x , x. : , . . . x p may be expressed in terms of the 
quantities v h — namely, they are the roots of an equation of degree p whose 
coefficients are products of d-functions of the v. If 
m. } ) = tm h m L a kh + 2tm h v;„ 
4 s 
MDCCCLXX Fill, 
4>{qn x , m 2 , 
