Functions given by Groups. 333 



questions about many-valued functions before the study 

 and formation of their groups, was to put the cart before 

 the horse. 



I have most convincing proofs that in the very highest 

 places of European science, this cart before the horse is 

 analytical orthodoxy. Again and again I have been told 

 by mathematicians of the foremost repute in foreign seats of 

 learning, where these subjects are really studied, that what 

 I have done in groups, in the Memoirs and Proceedings of 

 this Society, will be very useful when once we have learned 

 how to form the right functions. Functions before Groups 

 is the opinion in fashion ; one can then amuse one's self by 

 finding the groups. 



In English, I have never even indirectly heard one word 

 of any opinion on my handling of either groups or their 

 functions. 



The cart before the horse is certainly the correct method, 

 where you prefer pottering behind an orthodox wheel- 

 barrow to doing the work like an Englishman. 



16. The only result in these functions for n>^ that I 

 know of, as won in the path of genuine orthodoxy, is the 

 six-valued function of six letters given in the admirable 

 Algebre Supirieure of M. Serret, p. 515, Paris, 1854. Its 

 form is a product in one line of five similar factors, one of 

 which is {ab-\-cd-\-ef). 



The Academy, after this success in 1854 of one of their 

 most eminent members, were naturally desirous, in 1858, to 

 encourage further effort in the same orthodox direction. 

 That Algebre says nothing of groups. 



In this year, 1891, I have, for the first time, had the 

 courage to unpack and lay in readable order all the con- 

 tents of this hard little bale of M. Serret. I have found 

 in it eight functions, 



AA'A''BCDD'E. 



