22 REPORT—1904. 
constants is increased. The theorem in which he formulated the true 
state of the case will be enunciated later on, after a brief summary of 
some more of Riemann’s results. 
After showing that an algebraic function, s, with m infinities on the 
surface, is the root of an irreducible equation of the nth degree whose 
coefficients are integral functions of the mth degree in z, the complex 
variable—in other words, after showing that there is an equation, 
n m 
F (s, z)=0, which is so associated with the surface that as z moves over its 
whole extent, s is a one-valued function of its position with m simple 
infinities—Riemann considered the question of the determination of the 
‘branch-points’ of the surface from knowledge of the associated equation. 
The essential property of a branch-point is not only that two branches of 
the function coincide for a certain value of z, but that, as 2 moves round 
the branch point, these branches interchange their values.! This opens 
up the possibility that two branch-points may, as it were, destroy 
nm m 
each other by coincidence ; and, by expansion of F(s, z)=0 at a point, 
Riemann showed that true branch-points only occur when “* =0, but not 
s 
oF, or else when e =) Tel and OP FO?F _ ce ars h OF _ 
8 ie 
0 Os? Os. 02 08 
)F O?FO?F | /OF . oF \? 
ae mee 
when 
: 3 Os 
and aS =0, but oa “6 9 the branches do not interchange their 
values as z moves round the point. In the discussion of the r branch-points 
which may thus disappear, Riemann explicitly limits himself by assuming 
that when branch-points coincide it shall only be in pairs ; he thus rules 
OF _o OF _o ag roc) wire fee 
Daa” CLO sent 0s? 022 \ Osdz 
when three branch-points coincide, two of them destroying each other. 
This limitation is, however, quite unnecessary, and in the subsequent 
adaptation of his results for the purposes of higher plane curves it was 
not adhered to.? Riemann used his determination of the number of 
arbitrary constants in the expression of an algebraic function of the 
surface to show that every algebraic function s’ with m infinities, can be 
expressed as the quotient of two integral functions.? The number of 
arbitrary constants in such a quotient will be, as required, m—p+1, 
provided that both numerator and denominator vanish at the 7 points 
in which two branch-points destroy each other ; and this condition is 
also required in order to ensure that s’ should, in general, assume two 
different values at such a point, although s has only one value. Now 
out altogether the cases in which 
Ow ; : ae ; ' 
ay? the differential coefficient with respect to ~ of the integral of the first 
kind, is an algebraic function on the given surface, and it is infinite at the 
branch-points of the surface ; moreover 2 vanishes at the branch-points 
8 
of the surface and at the 7 other points as well ; lence the simplest 
w 
Oz 
_-; the numerator is then a function which Riemann denotes as 
expression as a quotient for is one in which the denominator is 
* Loe. cit. § 6. * See Clebsch, Crelle, vol. Ixiii. p. 192. ® Loe, cit. § 8. 
