GROUPS AND MANY-VALUED FUNCTIONS. 355 



of his own results in this first chapter. He has given 

 abundant evidence, in the subsequent parts of his Memoir, 

 that he is competent to deal with this dilQ&cult subject, 

 and has here simply repeated an oversight which probably 

 runs through previous French writers, — the oversight of 

 this principle : that in order to prove that a function of 

 N letters has K values, it is required not merely to shew 

 that it is invariable by the substitutions of a certain group 



JTN 



of the order -t^--, but also that it is variable for all other 



substitutions. 



I believe that the true theory of the connection between 

 groups and functions to oe constructed on them was first 

 given, and I hope completely given, in the tenth section 

 of the preceding Memoir (Art. 52). 



70. I have now to correct an error of my own. This 

 error is in the enumeration of equivalent groups in theo- 

 rem D (14) and theorem E (17). My mistake was in the 

 assumption, that no substitution made with N letters of 

 the form pi + c can ever be of the W^ order, for p>i. 

 This assumption is confuted by my own remark at the 

 end of Art. 19, page 293, that there are substitutions of 

 the eighth order (N = 8) of the form 5^ + ^?, viz. in the 

 derivate 5iG-. 



It is necessary here to determine the condition that the 

 substitution pi + c shall be of the W' order, where j9 < N, 

 > 1, is prime to N, and c is any of the numbers 012 • • 

 (N- 1), N being any number whatever. 



We have, (Art. 15), (let me beg the reader to prefix 15 

 to the twentieth line of page 287), 



{p)i-\-c)[p)i + c)[ = )pH ^pc ■\-c{ = ){pi-\-cY 



[pi + c) {pH +pc + c){ = )pH +p'^c +pc + c{ = ){pi + cf 



{pi + c){pi + cY'-^{ = )p"'i+p'"-^c+p'"'~^c+ • . +pc + c 



