8 
MR. E. T. WHITTAKER ON THE CONNEXION OF 
Let 
Let 
Then 
/ M 
V ’ 7 ./ + K) 
S 
i'+i — 
at + h\ 
ct — «/ 
and let S, 
s,. 
+1 
T, 
S, 
{go.,. + 57,,) t + + h g,.) \ 
(c«,. - «7d t + {cl3, — a S,)/ 
If this is a self-inverse substitution, we have 
a (a,. — 8,.) + + c/ 3 ,. = 0. 
Thus the coefficients of the substitution S^+i must satisfy the conditions 
(«! — 8i) a + yib + /3iC = 0" 
(a, — 82 ) Cl -j- 72^ d“ = 0 
. 
(%• ~ Si) + y^b + /3i,.c — 0 ^ 
The elimination of a :b : c, from these equations gives {k — 2) conditions between 
the coefficients of the substitutions T. 
[Added June 2, 1898 .—These conditions are sufficient, but are not actually neces¬ 
sary, as it may be possible to generate the group from a different set of substitutions, 
for which these conditions are satisfied, although they may not be satisfied by 
T.T2, ...T,.] . 
We shall, later, take k = 2 'p, and show that these (2p — 2) conditions must be 
satisfied by the coefficients of 2j9 substitutions, whose group gives rise to automorphic 
functions which uniformise a hyperelliptic form of genus p. 
§ 3 . The Division of the t-plane, corresponding to a group formed of Self-inverse 
Substitutions with Real Coefficients. 
A method will now be given for dividing the i-plane into regions, corresponding to 
a group generated from a given set of self-inverse substitutions. These regions are 
to be derivable from each other by applying the substitutions of the group. 
Let 
ct — a 
