841 



that on the substitution of 



y^=^, (2) 



•<■/ 



for ,vi, the constants of the transformed equation assume exactly the 

 same numerical values Sj as in the first case. We call such values 



ci'i", .i\", .... ,r„ corrfispondent to the values .u/, x^, .i-,,, and the 



state defined by the values xï' , correspondent to that defined by the 

 values X! (or corresponding to it). 



The form to which the given ecjuation is reduced in this case 



Xi Xi 



by the substitution yi = ^_, resp. i/i^~y^, will be indicated by the 



Xi Xi 



word universal. 



§ 3. When for the system x/ the system x/' corresponding to it 

 has been given, the system xn" can be easily calculated, which cor- 

 responds with every other system xn' of xi values, which satisfies 

 the equation in the first case, by the aid of the follow^ing equations: 



Xi^^xT^ 



I II ' 



Xil Xti 



Indeed the values Xi' resp. xi" satisfy the original equation, when 

 the parameters Q assume in it the values d' resp. Q". When now 

 the substitution 



!/i^^ (3) 



X ii 



has been carried out, the constants Si which w^e have calculated, 

 assume other values, e. g. Sn, and we must now find the values .^vi", 

 which keep the quantities Sii invariant on substitution of 6V' for 6'/', 

 when the substitution: 



Xi 



yi=—i (4) 



-Vil 



is carried out. 



The values x/, however, satisfying the given equation, 



ya — — , 



Xii 



satisfy the transformed equation. The constants of the transformed 

 equation do not change, when 



is substituted for yn'. 



