65 
This theorem might be extended. When n r - 2 l ~ 16, there 
is a noil-modular group made with 15 of the order 15T4T2’8, 
a non-modular made with 14 of 14'1 2'8, and a non-modular 
made with 1 2 of 1 2'8, which has a derived derangement, making 
it a modular group of \2'8’2. These are all transitive groups 
and easily constructed, either with or without the aid of the 
notation of linear substitutions made with 16 letters. We 
have here the true generalisation of the group of 7'4‘3'2'\ 
made with 7 letters, about which, with the 80-valued 
functions given by it, so much labour, far more learned 
than lucrative, has been spent by MM. Betti, Kroneker, 
Hermite, and myself. 
It is curious, that the true analytical expression of these 
transitive groups of n r - 1 and w’ - 2 letters is obtained by 
considering them as intransitive groups of n r letters. 
