Functions formed on Groups. 3 1 



If opycte be written in the paradigm over each of the 

 60 rectangles of 12 substitutions, and if they be then 

 read as 60 sums of 1 2 products of powers xfxfi . . Xo + 



being the two first of the 60 sums, the function on G + ' ■ is 

 correctly dictated with its 60 values under S = a/3y^ee. 



If we wish to dictate Gt 5 ii\ page 25, under 2, which is 

 (452613)+ •• under 2, given by its principal substitution, 

 whose vertical circle, read under I in the group of its six 

 powers, is 146325, we have to turn our G + •■ into Gtsd- 



We look for this vertical circle and find it in the 47th 

 rectangle of G + ' •, the derivate 146325c,! = fG d . 



We want 



G a6 g=fG d <l>- 1 = i46325G d i46325- 1 = 1463250^54263 . 



The dexter operation of _1 = 1 54263 on this47th rectangle 

 so permutes its entire and unaltered vertical rows, that they 

 become exactly G 25d headed by 123456; and, by writing 

 over this deranged rectangle our selected 2, we make it 

 (5 2 5rf- We have only thus to derange by <p~ x the other 59 

 rectangles, and to impose over them the same 2, and we 

 have plainly before us <$t 5d , with the function O ua under 

 «/3ySte in the 47th rectangle. 



For one of the 59 is G d) which so deranged has become 

 Gaty' 1 , = <p~ 1 G 25d , and is now a derivate of G 25d . Another 

 is BG,,, which has become 



BG,^- 1 = B<p-\$G d f- 1 ) = B^G^a, = CGaw, 

 another derivate of G 25d . And thus G d and the 59 different 



