ALGEBRAIC FDJICTIONS WITH AHTOMORPHIC FUNCTIONS. 
21 
We could have foreseen this by regarding the problem as one of conformal repre¬ 
sentation. The algebraic functions can be regarded as uniform functions on a 
Riemann surface which covers the 2-plane twice, the branch-points being at the 
points Cl, Co, . . . Now join the point e.^+2 to each of the points Ci, eo, Cg,. . . 
Then each of the sheets, regarded as an infinite plane bounded by these lines, is 
represented conformally on one of the [n -j- l)-gons in the f-plane; by taking two 
adjacent (r/, l)-gons, we obtain a 2n-gon, which corresponds to the fact that by 
taking the two 2-planes, and connecting them along the line e.^+2^n+i, we obtain the 
Riemann surface as dissected by n cross-cuts. 
The analytical connexion between the variables in a hyperelleptic form and the 
uniformising variable t is therefore given by the equations of § 5 . It can be shown 
that the differential equation found there is, as might he expected, one of Klein’s'^ 
“ unverzweigt differential equations for hyperelliptic forms. It can be obtained by 
equating (p — 2) of the arbitrary constants in Klein’s equation to zero. 
There are p integrals of the first kind connected with the form. It is easily 
proved that if v is one of them, then v undergoes a projective substitution of the 
form 
{v, c — v), 
where c is a constant, when t is transformed by one of the generating substitutions 
of the group. 
The theory of Abelian integrals of the form can be developed with t as independent 
variable ; but developments of this kind are outside the scope of this paper. 
One consequence of the results just obtained is that we can find the conditions that 
2 p arhitrarily given projective substitutions may generate the group corresponding to 
a hyperelliptic equation of genus p. 
Let the substitutions be Ti, R . . . Ih^, where 
rp _ (, 
’• ~ V’ + dj • 
On comparing the results of this section with those of § 2, we see that the condi¬ 
tions may he expressed in the form 
a^ dy. by Cy 
cq dg bg Cg 
di bi c^ 
[Added June 3 , 1898 .—These conditions are not, however, proved to be strictly 
necessary, since the group may be generated by another set of substitutions to which 
these conditions apply, although they do not apply to Th, T.,, . • . T2^. And the 
= 0 , (r, s, = 1 , 2 , 3 ,. . . 2 p). 
* ‘ Gofctinger Nacliricliten,’ 1890, p. 85. 
