141 
existence. It turns out, that the ingenious and learned 
efforts of the French and Italian mathematicians to conquer 
this theory by algebra, with its formidable army of congru- 
ences and imaginaries, have been from the beginning one 
brilliant error. The problem is tactical, and the solution is 
tactical. I have from the first suspected this, from the fact, 
that so very few substitutions have been shewn to be 
expressible by algebraic formulae, without a crowd of 
imaginaries, and that, with these few, operation and compu- 
tation are possible only at the cost of a most irksome 
complexity. My tactical methods handle at once the nume- 
rals (taken for the elements) which this vast algebra vainly 
attempts to symbolise, and all equivalent substitutions and 
groups, whatever the number of elements may be, are alike 
under control. And, what is the most important thing , the 
groups so constructed are exactly the many-valued functions 
desired. Having your group on the page, you have, in three 
more seconds, in its most simple and useful form for com- 
parison or for all computations, one of the explicit functions 
required. You write a single line of exponents over unity, 
attributing the same exponent to every element in the same 
vertical row, and your function is before you on the paper. 
If you choose your exponents properly, you have the simplest 
possible function which has the required number of values. 
All this I have already shewn, in the tenth section of that 
Memoir which was consigned to oblivion at Paris in 1860 ; 
and it has been nowhere else shewn. 
I believe that no transitive groups exist for values of 
nz.ll, which are not found in the following tables. „ If any 
have escaped me, it is the fault, not of my method, but of my 
carelessness ; and I hope that allowance will be made for the 
extent and difficulty of the negative which has to be proved. 
The tactical construction of all these groups is given in the 
Memoir here presented. 
