ALGEBRAIC FUNCTIONS WITH AUTOMORPHIC FUNCTIONS. 
15 
Thus dz/dt has values equal in magnitude, but opposite in sign, at the points 
t and t' ; and therefore, making t and t' to coalesce in the double point, dz/dt is zero 
at the double point. 
Let us now enumerate the constants at our disposal, in order to see the corre¬ 
spondence between the arrangement in the 2:-plane and the group of substitutions. 
The ^-figure is determined by n + 2 self-inverse substitutions. Si, S2, . . . S,j+2, 
satisfying the relation 
S1S2S3 . . . S„ + 2 = 1.(!)• 
There are three real constants, a, b, c, in each substitution, 
relation 
+ 6c = — 1, 
But by reason of the 
these three are only equivalent to two. Thus from the {n + 2) substitutions we 
get ( 2 ?^ + 4 ) real constants. 
The relation (1) defines three of these constants in terms of the rest. Also, this 
group is not essentially different from one which is obtained by transforming it with 
any real substitution, which shows that three more of the constants are non-essential. 
So there are altogether {2n — 2) essential real constants involved in the ^-figure. 
Now considering the 2;-plane, there are 7i 2 points Cj, 62, ... e,i ^21 and each of 
these is defined by two real co-ordinates, giving 2n 4 as the number of real 
constants. But we can make a homographic transformation of the plane, so as to 
transform any three of the points into three arbitrary points. This shows that 6 of 
the constants can be disregarded as non-essential. So we have (2n — 2) essential 
constants in the 2-figure. 
Hence the number of essenticd constants is the same in the z-fgure as in the 
t-figure. 
[Added June 2, 1898 ,—This does not in itself prove that for every 2-figure there 
exists a corresponding ^-figure; but the general existence-theorem of Poincare and 
Klein can be applied to complete the proof.] 
Hitherto we have derived the 2-figure from the t-figure. The next section is 
chiefly concerned with the converse problem of deriving the Cfigure from the 
2-figure. 
§ 5 . The Analytical Relations betiveen 2 and t. 
The analytical relations between 2 and t are of two kinds; (a) those which express 
2 ill terms of t and the constants of the substitutions, and {A) those which express 
t in terms of 2 and the quantities Ci, €2, .. . e.^ + 2- 
The Thetafuchsian series of Poincare solve the first problem for all classes of 
automorphic functions. We shall therefore only discuss relations of the kind (/ 3 ). 
As any quantity of the form {at + b)l{ct -fi cV), where a, 6, c, d, are arbitrary real 
