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groups ol H, by the method of Art. 89, except by taking for 
unity in (y) one of the seven forms following, 
1 2 2 3 4 5 6 7 8, 1 2 3 4-5 6 7 8, 1 2 3 4 5 6 2 7 8, 
1 2 3 4 5 6 7 8 2 , 1 2 2 3 4 2 5 6 7 8, 1 2 2 3 4 5 6 3 7 8, 
1 2 2 3 4 5 6 7 8 2 , 
and then affecting every element in (y) with the exponent 
read over it in unity The truth is that (y) is an effective 
auxiliary if we write it thus : 
1 2 3 4 5 6 7 8 
2 1 4 3 6 5 8 7 
3 4 1 2 7 2 8 2 5 2 6 2 
4 3 2 1 8 2 7 2 6 2 5 2 (g') 
5 6 7 2 8 2 1 2 3 2 4 2 
6 5 8 2 7 2 2 1 4 2 3 2 
7 2 8 2 5 6 3 4 1 2 2 2 
8 2 7 2 6 5 4 3 2 2 1 2 , 
which can he proved, as in Art. 39, to be a group equivalent 
to (g), and different as an auxiliary from the preceding eight. 
Seven mo.e auxiliaries can be formed as above from ( g '), and 
thus we can construct on (g), not eight only, but sixteen 
equivalent groups K of the 16th order with 16 letters, all 
having 15 principal substitutions of the second order, and 
all beginning with 
1234567890 ab cd e f 
21 43658709 la dc f e = 0, 
by the method of Art. 39. There are in all lSTSTl^'T'S'ST 
equivalent substitutions 0 : if we employ them all, and with 
them 16 variations of each of the 30 equivalents of ( g ), we 
shall construct 
^g(16-30T5T3Tl-9-7-5-3) = Q 
different equivalent groups K of 16. Dividing II (16) by 
this number of equivalents, we obtain for the order of the 
maximum modular, made by adding to K all its derived 
derangements (Theorem A, Cor.), 
16T5 ; 14T2-8, 
and this maximum modular has Q equivalents. 
