GROUPS AND MANY-VALUED FUNCTIONS. 351 



The function is (a/377SS = 321100) 



Y= 1^2234 + 5312^2 + 433365 + 2-Y3I + i'2"65 

 + 534261+632235+13522^4.^312624.33^2^1 



+ 331225 + 435216 + 634231 + 332254 + 631223 - 



+ 6V43 + 3'6^24 + 23i256 + 536234 + 63325i 



+ 2 Y16 + 134253 + 533226 + 234215 + 2^6214 



+ 435252 + 582213 + 135245 + 385262 + 584232 



+ 183264 + 285264 + 381264 + 284236 + 281243 



+ 585221 +3342i24-4'2^5i +6=^1^54+ 1V62 



+ 4^i"35 + .5^1^63 + 135223 + 4Y12 + 4^6231 



+ 5^2245 + 283245 + 583214 4. 33^2^6 + 2^5253 



+ 3862154.635312 + 584225 + 582241+533224 



+ 4^5213 + 482236 + I V36 + 1^3^52 4- 3^2215. 



The other four groups each give a value of the same 



function W of 120 terms of the same degree {a/ByySS 



= 321100). 



67. The system of exponents aajS/S'yy constructs the 

 same algebraic function on G^ and on (Artt. 57, 58) 

 6^214355 0^214355 G<i2 13455 0^124365 

 (^^123455 G^i24355 G^2i3456. 

 Of the six groups there are four which have the substi- 

 tution 214365^ and none of the four has any of the other 

 substitutions above written. 



Wherefore these four groups all give a value of the same 

 six- valued function of 60 termSj which is putting {aa^ ^<yy 

 = 221100), 



Z = i22'^34 + 2 Y45 + 324^5 1 + 6^25 4- 3'6'4c5 

 + 5^5^42 + 5^1^53 + 522^14 + I V53 + 6^5^34 

 + 3^1^25 + 4^223 1 + 5Y42 + 2^6^34 + 1^2245 

 + 425251+525212 + 1^5^23 + 521^23 + 421225 



+ 2232 1 5 + 32422 5 + 22 1 265 + 52423 I + 52g26 I 



4- 5Y63 4- 1^64 4- 4V12 + 325214 + 522251 

 + 324212 + 62i'»45 + 124236 + 422253 + 425225 

 + 123246 + 22325 1 + 522254 + 3252 1 4 + 322246 



