ALGEBRAIC FURCTIORS WITH AUTOMORPHIC FUNCTIONS. 
7 
Ill this exceptional case, an arbitrary constant, c, is introduced. Ihe reason is, 
that the quantities «! — 8i, — §2) Yu Ji-, all vanish, so the two equations for deter¬ 
mining CG '.h-i'.Ci reduce to the single equation 
C 3 = 0 . 
Any group of substitutions ivhicli is formed from (^ + l) self inverse substitutions 
as generating substitutions, alivays contains a self-conjugate sub-group luhich can be 
generated from h substitutions. 
For let G he a group formed from (Z; -{- 1) self-inverse substitutions Sj, S2, S3,... S^.+ i. 
Then, as before, any substitution of G can be written in the form 
Now let 
Then 
s = s,s s,,s,s,... s,. 
±1 - -L2 - ^/u + 1^^25 • • • -L/r - 
S,S, = SA-.iS..iS, = lf% 
Therefore, if the number of substitutions in S is even, % can be expressed in 
the form 
V _ T’-rp 'T'-rp '^p 
SO N is a substitution of the group generated from Tj, T2, . . . d\., 
If the number of substitutions in 2 is odd, we have, therefore. 
and as 
we have, in this case, 
2 = T;T^... T^S,, 
Q _ T-iQ 
10,. — i,. 
v* _ '■ps'T-iQ! 
^ — J. pX q • • • O/; q. 1 • 
So any substitution of the group G can be expressed either in the form 2 ^, or in the 
form 2^,Si+i, where 2^, is a substitution of the group H, which is formed from 
Ti, To, . . . Tj. And as in the case k = 2 , which has been already discussed, we see 
that H is a self-conjugate sub-group of G. 
[Added June 2, 1898 .—H may, of course, coincide with G; I am indebted to 
Professor Burnside for the example, 
Sr = 1, Si = 1, Si = 1, (S.S2S3)^ = 1, 
in which this happens.] 
To find the conditions that a group H, generated from any k arbitrary projective 
substitutions, Ti, To, . . . Tj., may in this way be a self-conjugate sub-group of a group 
G formed from (Z: -j- 1) self-inverse substitutions. 
