( 392 ) 



Gul of' the second equation (2) we still 11 nd 



1/7 dt 



or 



|/Q . (IT = \/q .dt=:dT (4) 



The iunclion t remains likewise invariant for the above mentioned 

 transformation apart from an irrele\'ant additive constant. 



The form / written as fnnction of t has therefore the form which 

 is independent of the chosen independent variable ; it will therefore 

 be characteristic for the connectioii existing- between two particnlar 

 integrals. 



All functions built up exclusively of / and t or deduced from 



dl dU r 



these, as — ,~r~,--, iTdr, etc. will be equallv proof against trans- 

 r/T dt* J 



formation of the independent \ariable. 



If we now choose r as independent variable then according to 



(4) f' =z\''(i, hence according to (2) Q=i and in consequence of 



(3) I =z2]\ hence P=z\l. The ditferenlial e(|nation (^1) assumes 



in this way the following standard form : 



d^x I ir) dx 



dx-^ 2 d,r < \ ) 



As in this equation only / and t appear, all invariant functions 

 (built up evidently out of the coefficients and the independent variable) 

 can be expressed in 1 and t. 



Let us now consider the so-called canonical form of the dilleren- 



dx 

 tial equation, i.e. the form in which the coefficient of — is zero: 



dt 



iC) 



ƒ pdt 

 c 



dt, ^= e ^ dt . . . {^) 



•-J' 



pdt 



q.-^q^^ (0) 



The expressions (5) and (6) for dt, and q, will be characteristic 

 for the connection existing between the particular integrals and will 

 therefore be invariant, that is 



