22 
MR. E. T. WHITTAKER OK THE COKKEXIOK OP 
inequalities expressing the conditions that the sides of the generating polygon do 
not cross must also he satisfied.] 
In all our work hitherto it has been assumed that p>l. Tlie case p = 1 is excep¬ 
tional; algebraic forms of genus unity cannot be uniformised by groups of the kind 
we have found. For if the construction which has been given were possible for 1, 
we should have, as the fundamental jDolygon of the group, a triangle whose sides are, 
in the non-Euclidian sense, straight lines, and the sum of whose angles is tt. But 
this is impossible, for in Lobatchewski’s geometry the sum of the angles of a 
triangle is always less than tt. When the sum is equal to tt we arrive at the limiting 
case of Euclidian geometry. Therefore the construction fails, and w^e have to devise 
instead a construction in which Euclidian geometry replaces non-Euclidian. We take 
four substitutions, So, S3, S4, satisfying the relation 
S,S2S3S4 = 1, 
which are self-inverse and leave the Euclidian absolute unchanged, i.e., w'hich are all 
of the type 
{t, c — t), 
where c is a complex constant. By reasoning exactly analogous to that in ^ 3 , we 
see that these substitutions generate a group, to which corresponds a division of the 
plane into rectilinear triangles. The sub-group which is got by taking adjacent 
triangles in pairs gives a division of the plane into parallelograms ; aiid this is the 
well-known group of the doubly-^iieriodic functions, which uniformise algebraic curves 
of genus unity. 
The following shows how the former construction breaks down in this case. 
If possible, let Sj, S2, S3, S4, be four self-inverse substitutions with real coefficients 
satisfying the relation 
S1S2S3S4 = 1. 
Then if 
S, = 
we have 
/ aj, -f 1}, \ 
V’ Crt - aJ ’ 
S1S2S3(0 
(aiCCoa^ 4- aJ)Xg -|- a^^h-^c., — t -j- -h 
(Ci«2®3 + <^\b-fz ~ t -f- {c^ctobs — cj).,as — — a^a.^a^ ’ 
This has to be a self-inverse substitution, since S4 is self-inverse. 
So 
aJo-Ps + ~ «2?>iC3 + epijbz — — apohs = 0, 
02 
O3 
h 
Cl 
C2 
C3 
