95 



17. There are three 6-valued equivalent functions, and no 

 more, of the 10th degree, made with six letters, each term 

 having four different exponents. 



18. There are two 6-valued functions, and no more, of the 

 7th degree, m-ade with six letters. 



19. There are two 6-valued functions, and no more, of the 

 6th degree, made with six letters. One of these is, putting 

 P2'3 4 for xixlx^oh, 



P223 4 -f snH 2 + 6-423 1 -f 22421 5 + 52224 6 



-I- 52426 1 -I- 62223 5 + 22125 6 -1- 62325 1 + 12326 4 

 -I- 32122 5 + 32622 4 + 52322 6 + 2262 1 4 -|- 52622 1 

 -f 625*4 3 + 12425 3 + 12524 5 4- 52423 2 + 32421 2 

 4- 22321 6 + 5222123 + 42126 2 -f 32524 1 + 52126 3 

 -I- 42525 2 + 42326 5 + 32225 4 -I- 62122 3 -f 42223 6 



which may be compared, as well as the preceding, with the 

 single 6-valued function of the lOth degree, discovered by 

 M. Serret, and given in his Algebre Superieure, Note viii. 

 This has, I think, been hitherto the French stock of 6-valued 

 functions, and we have repeatedly heard of it. 



20. There are 45 equivalent 45-valued functions, and no 

 more, of the I5th degree, made with six letters. 



21. There are nine, and only nine, equivalent 45-valued 

 functions, of eight terms, of the 10th degree, made with §ix 

 letters. 



22. There are eighteen, and only eighteen, equivalent 45- 

 valued functions, each of sixteen terms, of the 1 0th degree, 

 made with six letters. 



23. Thee are threre, and only three, equivalent 45-valued 

 functions, each of four terms, of the 7 th degree, made with six 

 letters. 



24. There are seven, and only seven, 45-valued equivalent 

 functions, each of eight terms, of the 7th degree, made with 

 six letters. 



25. There are seven, and only seven, 45-valued functions, 

 each of sixteen terms, of the 7th degree, made with six letters. 



