a 
ON THE THEORY OF POINT-GROUPS. oF 
published in 1851.' ‘This surface is an essential feature of Riemann’s 
method ; the results in the theory of functions which he obtained are all 
intimately connected with its use ; and it is as a consequence of this fact 
that the possibility of the transference of these results into the theory 
of higher plane curves naturally presented itself to his readers. A 
Riemann surface is, in its original state, ‘multiply connected,’ 2.¢., it 
cannot be bounded by one continuous curve ; its dissection into a 
‘simply connected’ surface, bounded by one continuous curve, is effected 
by means of ‘cross-cuts.’ In one of the opening sections of his ‘Theory 
of Abelian Functions’? Riemann takes the number of these cross-cuts 
for any given surface to be 2p, and this assumption provides the original 
definition of the number p, a number which reappears on every page of 
this memoir, and whose importance is marked in § 11 by the establish- 
ment of its permanence under bi-rational transformation of the equation 
F (s, z) = 0, associated with the given Riemann surface. In§$ 12 Riemann 
further pointed out that all algebraic equations can be divided into 
classes : those of the same class are derivable from each other by bi- 
rational transformation, and are characterised by the value of p. This 
classification is of great importance ; for subsequently, when the number p 
has been identified with a purely geometrical property of a curve, it is 
shown that, starting from this as a definition, p is permanent under a 
bi-rational transformation of the curve ; and the standpoint of projective 
geometry, from which curves are classified according to their orders, is 
almost entirely replaced in the theory of point-groups by the standpoint 
of bi-rational transformation, in which a curve is classified according to 
the value of p. 
Riemann regarded all the functions with which he dealt not, as other 
writers had done, from the point of view of the actual functional form 
they possess, but as defined by certain properties? Among these 
properties is the existence of infinities of given orders at given points 
of the Riemann surface, and of given ‘moduli of periodicity’ at the 
2p cross-cuts. The simplest case is that in which the function has no 
infinities, but only moduli of periodicity : these were called by Riemann 
‘integrals of the first kind,’ and, from the fact that there are 2p cross- 
cuts, he showed that there are exactly p linearly independent integrals 
of this kind on a surface.t He next discussed integrals of the 
‘second’ and of the ‘third’ kinds, which have respectively algebraic 
and logarithmic infinities, and then proceeded to form an algebraic 
function by means of a sum of integrals of the first and second kinds, 
together with an additive constant, this sum being subject to the 
condition that the moduli of periodicity should vanish.® And, by 
counting the constants in the equations which express this condition, 
he found that, after the m infinities of an algebraic function have been 
chosen on the surface, there will always remain precisely m—p-+1 
arbitrary constants (including the additive constant) in its expression.® 
This result was corrected seven years Jater by Roch (1#39-1866), 
who pointed out that under certain conditions the number of arbitrary 
! «Grundlagen fiir eine allgemeine Theorie der Functionen einer verianderlichen 
complexen Grosse,’ Inaugural- Dissertation, Gottingen, 1851, Ges. Werke, pp. 3-47. 
2 «Theorie der Abel’schen Functionen,’ Crelle, vol. liv., 1857; Ges. Werke, 
2nd edit. 1892, pp. 88-144. 
3 Loe. cit. § 3. 4 Loe. cit. § 4. 
5 Loe. cit. § 5. § Loe. cit. § 5, 
