6 
MR. E. T. WHITTAKER ON THE CONNEXION OF 
But 
Q Q _ '•p-l'T' Q Q _ T-' 
^1^2 — -Ai -A-J, D1O3 — i, , 
QQ _ 'T-l QQ _ 'T' 
^2^53 - ±2 , ^3^1 — -Ai, 
5351 TT^Tj, 
5352 = T,. 
Therefore every yjair S,,S, can be expressed in terms of Tj and To. 
So if the number of substitutions in % is even, the whole substitution can be 
expressed in the form 
V = TrrfTjTI . . • 
i.e., it is a substitution of the group H. 
But if the number of substitutions in S is odd, there will be one substitution S,, 
left at tlie end unpaired. Now 
so m any case 
s. = Tr’S3, S2 - T2-tS3, S3 - S3, 
S = TtTfTiT ?2 . . . Tr.S.3. 
So is always eitlier a sul)stitution of H, or else the ])roduct of S3 and a substitution 
of H. 
Now let S/i be any substitution of H, and S,, any substitution of G. 
Then Sy^S/,S^ evidently contains, when decomposed into the substitutions S,, So, S3, 
an even number of them ; for S/^ contains an even number, and Sy' and Sj, each 
contain the same number. Therefore Sy’S/,S„ is a substitution of the group IT ; which 
establishes the required result, namely, that H is a self-conjugate sub-group. 
As an example of this theorem, consider tlie modular group generated by the 
substitutions 
(b ^ + 1) and (^b — 
This is a self-conjugate sub-group of the group formed from the three self-inverse 
substitutions 
- 1), (b-), (b - t). 
As another example, take the group which occurs in the theory of elliptic functions, 
which is formed from the generating ’distitutions 
(b t 4- 2 iv^), (b t + 2 w.,). 
This is a self-conjugate sub-group of the group formed from the three self-inverse 
substitutions 
(b c — 2'«Cj — t), (b c — 2 u) 2 — t), (b c — t) 
where c is an arbitrary constant. 
