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, made with six letters. 
19. There are two 6-valued functions, and no more, of the 
6 th degree, made with six letters. One of these is, putting 
P2 2 3 4 for xjxlxsCCi, 
1 2 2 2 3 4 + 5 2 1 2 4 2 + 6 2 4 2 3 1 + 2 2 4 2 1 5 + 5 2 2 2 4 6 
+ 5 2 4 2 6 1 + 6 2 2 2 3 5 + 2 3 1 2 5 6 + 6 2 3 2 5 1 + 1 2 3 2 G 4 
-1- 3 2 1 2 2 5 + 3 2 6 2 2 4 + 5* 3" 2 6 + 2 2 6 2 1 4 + 5 2 G 2 2 1 
+ 6 2 5*4 3 + 1 2 4 2 5 3 + 1 2 6 2 4 5 + 5 2 4 2 3 2 + 3 2 4 2 1 2 
+ 2 2 3 2 1 6 + 5 2 2 2 1 2 3 + 4 2 1 2 G 2 + 3 2 5 2 4 1 + 5 2 1 2 G 3 
4- 4 2 6 2 5 2 + 4 2 3 2 6 5 + 3 2 2 2 5 4 + 6 2 1 2 2 3 + 4 2 2 2 3 6 
which may be compared, as well as the preceding, with the 
single 6-valued function of the lOtli 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 15th 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 six 
letters. 
22. There are eighteen, and only eighteen, equivalent 45- 
valued functions, each of sixteen terms, of the 10th degree, 
made with six letters. 
23. Thee are threre, and only three, equivalent 45-valued 
functions, each of four terms, of the 7th 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. 
