| 
ON THE THEORY OF POINT-GROUPS. 93 
n=—2 m—2 
@ (s , 2), and it also must vanish at the 7 points.'| This function ¢ has, 
in general (n —1) (m—1)—r arbitrary constants, a number which Riemann 
had previously shown? to be equal to p. There are, therefore, p linearly 
independent functions ¢, which agrees with the former statement 
that there are p linearly independent integrals of the first kind. The 
function 9, which thus appears quite naturally in this expression for 
w 
dz” 
through its influence on the theory of higher plane curves, is connected 
with the theory of point-groups. It is characterised by the fact that it 
must vanish at the 7-fixed points on the surface where a pair of branch- 
points destroy each other ; moreover, as Riemann went on to show, it 
has also m(n—2)+n(m—2)—2r=2p—2 other zeros which may be any- 
where on the surface ; * in contradistinction to the fixed zeros, common to 
all »’s, these latter are now often spoken of as the moveable zeros of a 
o-function. Roch’s correction of Riemann’s enumeration of the number 
of arbitrary constants involved in the expression of an algebraic function 
on the surface—alluded to above—is concerned with this function », and 
his theorem is as follows:* ‘ If an algebraic function s’ have m infinities 
plays an important part in that portion of Riemann’s work which, 
nm 1b 
on the surface associated with F(s, z)=0, and if q functions ¢, which 
are linearly independent of each other, vanish in these m points, then 
s has m—p+1+¢ arbitrary constants in its expression.’ 
This is the theorem known in the theory of functions as the Riemann- 
Roch theorem ; transferred into the theory of higher plane curves it 
became part of a more general theorem, now usually spoken of as Brill 
and Noether’s Theorem of Reciprocity. The latter, which is of funda- 
mental importance for the theory of point-groups, was not taken from 
any theorem which had been explicitly stated in the theory of functions, 
but its enunciation was suggested by certain results obtained by Riemann 
in connection with the theta-functions.® These results are based upon the 
discussion of Abel’s theorem, which occupies the last three sections of the 
first part of Riemann’s ‘Theory of Abelian Functions.’ Part II. of this 
work is devoted to the theta-functions, but the actual results in which we 
are interested appear in a later memoir (1865) on the vanishing of the 
theta-functions.“ A short account of these investigations will now be 
given in order to show how the first idea of a point-group arose. 
Riemann was only directly concerned with Abel’s theorem in so 
far as it applied to integrals of the first kind; he was in fact the first 
definitely to enunciate and prove it for this case, for in Abel’s original 
discussion of the subject this case had been treated as a particular instance 
of the more general theorem for the three kinds of Abelian integrals, and 
the conditions under which the sum of the integrals reduces to a constant 
(i.e., the case in question) are complicated, and involve possibilities which 
are excluded by Riemann’s method of attacking the problem. This 
method, moreover, is readily applicable to the other cases, which, however, 
do not concern us here. The ¢-function, which enters, as we have seen, 
' Loe. cit. § 9. * Loe. cit. § 7. 3 Loe. cit. § 10. 
* Roch, ‘Ueber die Anzahl der willkiirlichen Constanten in algebraischen 
Functionen,’ Crelle, vol. lxiv. pp. 372-376 (1865). 
®> See Brill and Noether’s Bericht, also Mathematische Annalen, vol. vii. p. 280. 
6 Riemann, ‘ Ueber das Verschwinden der Thetafunctionen, Crelle, vol. Ixv. 
1866; Ges. Werke, pp. 212-224. 
