352 REV. T, p. KIRKMAN ON THE THEORY OF 



+ a^^^gg + 52321 2 + I Y25 + I Y65 + 5'i'62 

 + 6^23 + 4V23 + ,5^6231 + 2^3 1 + 221^36 

 + 5^1234 + 331^42 + 422^16 + 126^42 + 4V6 1 

 + 422^65 + 226^53 + 3V62 4. ^52^2 j^ 4_ ^2^25^^ 



The other two groups have the substitutions 

 124365 213465 214356, 

 and they, consequently, each give a value of the same six- 

 valued function U of 30 terms of the degree {aa/3^yy 

 = 221100). 



This function is 



U=i22234 + 53i242 4-6-'423i + 2¥i5 + c5-2^46 

 4- 5V61 + 6-2^35 4- 221^56 4- 633251 4- 123364 

 + 321225 + 326^24 + 523226 4- 226214 + 52622 1 



+ 6^43 + 1^53 + i'6^'45 + 524232 4- 3¥i2 

 4-233216 + 5222134-421262 + 3252414-52,263 



+ 425252 + 423265 + 332254 + 62l323 + 4^2236. 



Supplement. 



68. The preceding investigations and theorems are 

 translated, except Art. (39) since added, from a larger 

 work, which I last year presented to the Imperial Insti- 

 tute of France, in competition for their Grmid Prix de 

 Mathematiques for the year i860, of which the subject 

 was proposed early in 1858 : 



" Quels peuvent etre les nombres de valeurs des fonctions 

 Men definies qui contiennent un nombre donne de lettres, et 

 comment peut on former les fonctions pour lesquelles il 

 existe un nombre donne de valeurs?" 



Three Memoirs were presented, but no prize was 

 awarded. Not the briefest summary was vouchsafed of 

 what the competitors had added to science, although it 



