S5i> Mr. G. B. Jerrard's Rejections on the Resolution 



which quantities are different from the former, but equal to 

 Ihem in number. 



If we operate in the same manner with (*^ )' ( A? )' 



, , I A ^ ) until we have exhausted all the substitutions, 



\-^(«-iV+i/ 



we shall find that the v values of X will be separated into co 

 groups composed each of them of jw, terms. 



Hence the number of different values which a function of 

 n quantities may receive from all the possible substitutions of 

 these quantities among themselves, is necessarily a submultiple 

 of the product I . 2 . 3 . . w, as is well known. 



16. If X be affected by a series of contiguous^ substitutions, 



(t)' (t)' a:)' •• (a:-)- 



we shall have generally 



This is evident. 



- 17. Let us now consider 



-©tt)(A:)- 



where the same substitution is supposed to be applied any 

 number of times in succession to the function X. 



It is obvious that a limited number, jo, of such operations 

 must bring us to an expression equal to X; and that all the 

 expressions previously obtained will then reappear in a perio- 

 dical manner. 



If, in effect, we denote by X ( . M the value of X which 



will arise when the substitution designated by ( * M has been 

 applied r times ; we shall have 



/A}\p 

 after which we shall come to the term X ( . I which, by 



/A \*' 

 hypothesis, is eqiial to X or X I .M ; and consequently if 



• A term made use of in connexion with substitutions by M. Caucliy, to 

 whom we are indebted for the results in (17.) and (18.), as well as for the 

 theorem given above in (16.). 



