4 
MR. E. T. WHITTAKER OK THE COKKEXIOK OF 
same as Klein’s “ unverzweigt ” differential equation for hyperelliptic forms, save 
that a number of constants left arbitrary in Klein’s equation are found to be zero. 
The conditions that 2/9 arbitrarily given substitutions may generate the group corres¬ 
ponding to a hyperelliptic equation of genus jj are found. 
In § 7, the consideration of the constants left undetermined in the differential 
equation of § 5 is resumed. If an algebraic form of genus p be given, the uniformising 
variable is one of variables, which are here termed “ quasi-imiformising.” Any 
quasi-uniformising variable affords a solution of the problem of conformally repre¬ 
senting the Ptiemann surface of the form on a plane area vPose sides are derived 
from each other in pairs by projective substitutions. The differential equations 
connecting the uniformising witli the quasi-uniformising variables of a given algebraic 
foi'in are obtained. 
§ 2. Properties of Self-inverse Substitutions. 
A projective substitution of a variable t is denoted by 
where we can always suppose that ad — 6c = 1. 
The substitutions, from which the groups considered in this paper are generated, 
are such that 
ct -j- f/ = 0. 
Such a substitution is elliptic and of period two ; its multiplier is — 1, and it is its 
own inverse substitution. For brevity we shall call such substitutions self-inverse.” 
Thus, if S denotes any self-inverse substitution, we have 
S“ = 1, and S = S-I 
If T he any substitution, and S be a self-inverse substitution, then T“^ST is a self¬ 
inverse substitution. For the multiplier of a substitution is unaffected by the trans¬ 
formation which chano’es S into T“'ST. 
If there be any number of self-inverse substitutions, and a, substitution be formed 
from them, then the substitution inverse to this is formed by taking the same substitu¬ 
tions in the reverse order. For if S^,, S,, . . ., are self-inverse substitutions, then 
obviously 
So if 
then 
. . . S„S,S„.S,,S,S.S,S,S, = 1 
T =:• S,S,... S„S.S„, 
T-' = SA.B.... S,S,S,, 
