GROUPS AND MANY-VALUED FUNCTIONS. 347 



function with its derangement ^^12354, which is the de- 

 rived of another group. 



The three functions are {a^<ySS = 22ioo) 



¥1=1^3 + 2^1 + 3^ 1^4 + 4^3^2 

 2^3% + 4^ 1^3 + 1^4-2 + 3^2^ 1 



4^5^ 1 + 3^5^2 + 2^5^3 + i^5^4 

 5=^1^-2 + 5^-4 + 5^3^1 +5-^3 



¥3= 1=^2^3 + 2^523 + 4^123 + 5V3 

 2^1+5^1^2 + i^5^4 + 4^2^5 

 4^2 + 1 V5 + 5''2^i + 2^i^4 



5^324 + 43321 + 13325 + l3g22 



3^125 + 3^224 + 33422 + 33521 



V3= 132^3 + 2^324 + 53122 -I- 33(|2i 

 23524 + 33^2^ _|_ J 3^3^ _|_ ^^2\ 

 53321 + 13522 + 38325 + 23i23 

 33422 + 53423 + 23421 + 13425 

 43125 + 48221 + 43523 + 43322 



The eighteen values of these three functions are all dif- 

 ferent. 



63. Let the system of exponents be a^^^yy. 



Two of the six groups give values of the six-valued 



function 



V4= 1^23 + 2253 



2241+5212 



4252 + 1245 



5'34 + 4'3i 

 3215 + 3224. 



The other four groups give values of the six- valued function 



^5= 1^23 + 2241 +3214 + 4232 

 + 2234 + 421 3 + 1 242 + 322 1 



+ 3'45 + i'35 + 4^25 + 2^15 

 + 4251 + 3252 + 2253 + 1254 

 + 5212 + 5224 + 5231 + 5243 



