24 
MR. E. T. WHITTAKER OK THE COKKEXIOK OF 
place occupied by branch-points at which only two branches interchange, in the 
theory of Riernann surfaces ; the usefulness of the method of self-inverse substi¬ 
tutions depends on the fact that algebraic forms can be represented on Pdemann 
surfaces with only simple branch-points. 
Algebraic functions are not, however, the only ones which can be uniformised. 
Poincahe"^ has proved a general existence-theorem that, if Ui, u^, . . . are any 
multiform analytical functions of a variable 2, a variable t always exists, such that 
2, Uy, U2, . . . u,,^, are uniform functions of t. The existence-theorem, however, does 
not connect t analytically with the other variables. If Ui, U2, . . . u^, are transcen¬ 
dental functions of 2, their multiformity will not in general be capable of beiug 
expressed by simple branch-points, and so the groups generated by self-inverse 
substitutions cannot be used. 
§ 7 . The Undetermined Constants in the Differential Equation connecting 2 and t. 
In § 5 , certain constants Jci, E, . . . in the differential equation connecting 
2 and t, were left undetermined. It was there explained that they are to be 
determined by the consideration that the group of substitutions of t leaves unchanged 
a fundamental circle. In general, however, arbitrary constants occurring in similar 
differential equations cannot be determined by this consideration, as the group may 
be “ Kleinian,” i.e., it may not conserve a fundamental circle. The following dis¬ 
cussion approaches the subject from this more general point of view. 
The Riemann surface, corresponding to the algebraic form f{u, 2) = 0, can be 
made simply-connected by drawing 2p cuts, and the problem of finding the uni- 
formising variable t can be divided into two parts, as follows : — 
1. Finding all the variables r, which are such that the dissected Riemann surface 
is represented on the r-plane by a curvilinear polygon, whose iqj sides can be 
derived from each other in pairs by projective substitutions of r. 
2 . Selecting from among these variables r, a variable t, which is such that the 
group generated from these projective substitutions is a discontinuous group. 
We shall call the variables r quasi-uniformising variables, to distinguish them 
from the true uniformising variable t. 
In the case of the groups we have found, the differential equation of ^ 5 gives 
the quasi-uniformising variables; the determination of E, . . . is equivcdent 
to selecting the uniformising variable from among them. 
In this section the connexion between the uniformising and quasi-uniformising 
variables is considered for more general groups. 
As an example of the nature of quasi-uniformising variables, take the algebraic 
equation 
= 42® — g^z — P' 3 . 
* ‘Bulletin de la Societe Math, de France,’ 1883, vol. 11, p. 112. 
