Mr. J. Cockle on Multiplicity of Values, 445 



where 



*i(y)=«i(y) x«i(y)> <*&)** *hM* € sM)>*> 



<z m {y)=a m {y)xe m (y), 

 and a and e are functions of one or more of the n quantities 



PiiSforiEifa 



III. Suppose that any one of the functions, say a (y), can be 



derived from any other of them, say a^y), by some change which 

 we may represent* by 



/I, 2,.,7l\ 



then I apply to the functions a the term commetric. 



IV. When the functions a(y) and e(?/) are symmetric, and the 

 functions a{y), e(y), and «(y) are commetric, I term &(y), if it 

 be not symmetric, an epimetric function. 



V. Let the function a(y) have only m values, then I call se(y) 

 a pure epimetric function ; and the application to it of any change 

 will produce another pure epimetric function, in which the values 

 of a{y) will recur in a different order. 



VI. We may in such case consider the functions a(y) and 

 their order as permanent, and, without regarding the portion a 

 of se, confine our attention to the changes which can take place 

 in the portion e } and which we may term the epimetric inter- 

 changes. 



VII. By an interchange I mean an operation which introduces 

 no new value of y into a function. I denote it by the peculiar 

 brackets {}. An epimetric interchange, which operates on the 

 functions e successively, I represent by e{\. Among the number 

 of interchanges (or rather of interchangeable values) the primitive 

 function is included. 



VIII. The number of values which a pure epimetric function 

 can take is a function of the number of epimetric interchanges. 

 In other words, if we denote by/(se) the number of values of 

 vd(y), supposed pure, and by e r the number of interchangeable 

 values of e r (y), we have 



/(*)=«Me)=ef(e), 



e r being the same for all values of r, and ^(e) representing the 

 number of values which, e r remaining unchanged, the residue of 

 the function can take. 



IX. Without seeking now to determine <f> in all its generality, 

 let us consider the function U 4 which occurs in my application 

 of the method of symmetric products to equations of the fifth 

 degree (Phil. Mag. for March 1853, p. 174). 



X. If in place of U 4 we write — 5w, and make 

 • 



* Phil. Mag. S. 3. vol. xxvi. pp. 552 et seq. 



