ALGEBRAIC FUNCTIONS WITH AUTOMORPHIC FUNCTIONS. 
5 
The group formed hy the combination and rep)etitio7i of any tivo p>rojective substitu¬ 
tions can be obtained as a self-conjugate sub-group of a group generated by three self¬ 
inverse substitutions. 
For let 
(f 
^ ’ i.t + s, / 
and 
u.y + (3-1 
id + \ / 
be the given substitutions ; let 
S, - t, 
('d + 
cy. 
^2 = b 
a.y + 1)., 
Cot ((o 
a.y + h.. 
’ cJ — «... 
be three self-inverse substitutions ; then we have 
and 
(aiC(.. + fibi) t -f (ftAi — afi.p 
- a.fP) t + {a^Uo + h^cl) 
{a./Cs + ^ + (»-A — 
(aoC-i — t + + bed ■ 
The equations to be satisfied by the coefficients of S3, in order that we may have 
reduce to 
SgS, = T,, and S3S2 = T9, 
(«! — 81) «3 fi- yfi + ^iCg = 0 
(a.2 — 82) fflg 'y.fs + A2C3 = 0 
These equations always admit of a solution for the ratios ag : 63 : C3, if the substitu¬ 
tions Tj and To are distinct. Thus, the substitution S 3 is determinate ; and then S, 
and So can be uniquely determined from the equations 
S, = S3T., S.2 = S3T.2. 
[Added June 2. In view of the subsequent limitations to substitutions for which 
-b be is negative, it should be noticed that these equations may give either a positive 
or a negative value for cd -f- 6c.] 
Now let G denote the group formed from the generating substitutions Sj, So, S3, 
and let H denote the group formed from the generating substitutions Tj and To. 
As Ti and T2 are themselves substitutions of the group G, the group H will be 
either the same as G, or a sub-group of it. We shall now show that H, is a self¬ 
conjugate sub-group of G. 
Since S^ = 1, and S^. = S, 7 b any substitution of G can be represented in the form 
S = S^S.^Sj.S,. . . S,,, v/here 2^’ q, r, s, . . . v = 1, 2, 3 . 
