S56 Mr. G. B. Jeii'ard few a Method of Transforming Equations 

 as there are different solutions, in positive integers, of the equation 

 a-^b-\- .. -\-l^n. *"' 'V.vv.ri"* 

 e»-^'::1i''Kii\ii' symmetric functioii^nm'*^^^!^^^- 

 xa:\,,.x^,,"^bX . . is a term. As for the number of terms in 

 Q)ot*p . . X', it is wholly unaffected by the equalization of the 

 elements of that function, u, a, . . ^, ^, . .\,X, . . . For example, 

 if we take the very simple function of the 2nd order @a/3, sup- 

 posing it to be symmetric with respect to two quantities t and u, 

 we shall have 



from which, on putting /Ssaa, there will be derived >^qx.9oriT 



@a' is thus composed of as many terms as @«y8. And a similar 

 mode of derivation is supposed to extend to eveiy order in th6; 

 present system of symmetric functions*. ...,lj, 



2. It is not difficult to demonstrate the truth of equation (a). 



In effect, from a well-known property of equations, the coeffi- 

 cient of y"»-» in the transformed equation in y will be at once 

 seen to be expressible by '^ "***? '*&' 



V ^ 1 .2 . .71 ^y^y^y^ --yn, i ^^f^.,^ ^ri wovr 



if we use the symbol S in an equally extended sense with @, 

 that is to say, if we suppose ^y^y^Vs "Vn to be derivable from 

 ^y^iVaVl • • Vn without any diminution in the number of its terms, 

 on taking a=b=c= ..=/=!. 



But, according to equation («), the coefficient of y"*~'» in tfie 

 transformed equation will be 



(-irY^©.(«p+iSQ-i-7R+..+^Lr; 



n being any integer less than m-4- 1. 



^"Nothing therefore remains but to show, that if "i^'^' 



I-'''' 'fi. 

 where t may have any one indifferently of them values l,2,3..m 



* In the systems of symmetric functions hitherto in use among mathe- 

 maticians, i*u*, not t*u*-\-u'^t*, would in the case we have been consider- 

 ing have the same characteristic as <*m^+m*^/^. But had we thus insu- 

 lated those symmetric functions in which there are equal elements, we 

 should not have arrived at a theorem in which the separation of the symbol 

 (& from its subjects could have taken place. 



