842 

 The fraction : 



Xl Xi 



Xii" Xil' 



belongs therefore to the corresponding values .a", hence -r,,' corre- 

 sponds to .Vi[''. 



Hence it is proved that in the case of a system of vohws corre- 

 sponding to a system of solutions, there also e.rists a sys/tm corre- 

 sponding to every other system of solutions (when C'/ ha\e been 

 replaced bj G"). 



§ 4. To find a system .Vi", if the system d',' has been gi\en, we 

 take into account every product of powers of the variables: 



Fi^z.r^an xf^'-i x,,ain (6) 



which appears as separate argument in the given equation. We 

 shall therefore write the given equation as follows: 



<P{K,P,,K\P, K,,Pj,;L,,L,....Lt) = 0. . . (7) 



Ki and Li are constants with relation to .r,, L; are those constants 

 which do not occur as factor of P, , but in any other way. Among 

 the Ki and Li are therefore also included the variable parameters 

 (for their functions). 



Let us put that the constants Ki , Li in the first, resp. second case 

 have the special values : 



AV, L/', resp. Ki",Li" 

 (those among them which are independent of Q , have the same 

 values in both cases); they are to be considered as given. We can 

 write every \ariable also in tiie following way : 



. _ .'-'''' _ ..' 

 Xi — Xi — Xi yi . 



If we put them in this form in the equation (7), it assumes the 

 following form : 



<P{Q,'P,{y),....Qf:Pk{y);L,' Li') = . . . (8) 



in which 



Qi' =Ki'Pi' (9) 



Pi' = .t-/«'i .i-;«r2 .... ..;"'■« (10) ' 



^dy) -= i/x"'^ J/."'- . • . . ^«"'" (11) 



Now it is evidently the question to tind such values .tv" that when 

 Ci' is replaced by d" and .r,' by Xi", all the constants Qi and Li 

 — eventually with the exception of one factor, by which all the 

 terms of the equation can be divided — assume the same values. 



