G78 
PROFESSOR W. K. CLIFFORD OR THE CLASSIFICATION OF LOCI. 
derived points, only certain forms present themselves. Let that number which, with 
k — 1 other numbers, makes up zero, be called the residual of those numbers ; it is, in 
fact, their sum taken negatively. Then the process of forming the group is to start 
from unity, and add the residual of every k — 1 numbers of the group, repetitions 
being allowed. I say that by this process we shall only get numbers of the form 
mk -f- 1 . For let m y k + 1 , mJk-\- 1, &c. he k— 1 such numbers, then their residual is 
— K+ m 3 + • • )k — k — 1, which is a number of the same form. Now as unity, 
with which we start, is of this form, it follows that all the numbers of the group must 
be of the form mk + 1 . — January, 1879.] 
CURVES OF DEFICIENCY p. 
Theorems relating to Abelian Functions. 
It will be convenient to put together shortly those propositions relating to the 
application of Abelian functions to curves which will be wanted in the sequel. 
The aggregate of the real and imaginary points on a curve constitutes a two-way 
spread, or surface, which may be transformed, by stretching without tearing, into the 
surface of a body with p holes in it. On this surface there are 2 p distinct closed 
curves which cannot without breaking be shrunk into a point, namely, one round each 
hole, and one through each hole. Any other irreducible circuit must be made up of 
combinations of these. 
If any rational function of the coordinates be integrated from one point to another 
along the curve-spread, the value of the integral will depend upon the path of the 
integration. If the integral becomes infinite at any points, it may be altered in value 
by making the path go round one or more of these ; but in any case it may be altered 
by incorporating into the path any of the 2 p closed circuits just mentioned. It is 
found that there are p distinct rational functions of the coordinates whose integrals 
do not become infinite at any point of the curve-spread. Any other integral which is 
everywhere finite must be a linear combination of these. Of such linear combinations 
it is convenient to take a certain set as the normal set ; they are so chosen that each 
of them, when integrated along a closed path which goes round a hole, gives zero for 
all the holes but one, and ui for that one ; thus, the p integrals, which we may call 
u y , u 2 , u 5 , . . . u p , are associated one by one with the p holes 1 , 2 , p. If they are 
integrated along a closed curve passing through the hole h, let the values be called 
ah i, U/, 2 , a ] lp ; then it is found that a^—a^, or the integral of ii/ t through the hole k is 
equal to the integral of Uj c through the hole h. 
If we now take all the integrals from a point a? to a point y along the same path, 
and if u x , %, . . . u p are the set of values for one such path, and U], U 2 , . . . for 
another path, then we must have 
