Functions given by Groups. 335 



with demonstration the work required. And to-day, heretic 

 that I am, I actually believe that I had done it all. 

 The Academy have never denied this. But the Academy 

 were not content. Would you know why ? Well, look at 

 the shocking way in which I did it all ; so utterly destitute 

 even of the vulgar grace of orthodoxy ! Nowhere — nowhere 

 was my cart ever seen trotting before my horse ! 



17. To sum up. For this problem of many-valued 

 functions we require the necessary data^ and the necessary 

 algebra. 



We have the first when there is before us a table (A„) of 

 Q equivalent maximum groups GiGa . . . Gq made with ;/ 

 elements, with all their derivates. 



By writing over the Q columns of each line any the 

 same selected S, we turn them into CBi"^' *, (^2"^ ' •,...(3q'^ "> Q 

 functions with all their values. It is easy by art. 8 to 

 reduce the Q functions in (H„) to r< Q, which have all 

 under the chosen S Q values. We have only to determine 

 how many of these r are distinct functions (Bi"^ * '■, &c. 



Form the index-group I^+i given by 2. Take any one 

 Gd of the r groups, and to the m + i substitutions {inyO) in 

 I^+i which are also in G^^, give in I^^^i the names i, 61, 02, . . . 

 0„i ; naming the other t-in substitutions of It+i, 0i, 62, . . . 

 Qt-m- Let 0;, stand for any one of these t - m. 



The algebra is completed by — 



where (Bd^x, 0r^(3A> ^xO/, stand for nothing but what GAj 

 0x~'GA, GxGf, become under S. 



We find Gf in our table (A,J and mark it out by a dy as 

 useless, because the value 6^,0 f is identical algebraically with 

 Od- And we have t-m such outmarkings with ^ to per- 

 form, one for every d^. We have then done with G^ : (3^ is 

 one of our sought functions. 



We take next any group G^^ in (A«) which is not marked 



