( o^^-t ) 



Instead of / and t we miglit luive taken ^y, and f^ as standard 

 functions. We find tlie expressions for J and r in r/, and f^ most 

 easily out of (3) and (4). Tiiese expressions fnrnisli 



dT=\/q,.<lt, (9) 



'=^^^t '''' 



If we now" start from tiie differential ecpiation given in the general 

 form (A) then the definition of / as function of t does not make an 

 integration necessary, whilst to get t as function of / we have 

 but to integrate once. The formation of / as function ofrfWyelima- 

 nation of t) requires therefore but one integration. 



The construction of <J■^ as function of /, requires on the contrary 

 (see (5) and (6)) two integrations. It will therefore be in general 

 easier to find /(t) than ^i(^i). 



Let us now suppose a relation between two particular integrals, 

 we can then make it our task to find the form of Pr) or of ^i(^,)- 

 It will then often be easier to detern)ine /(t) than q^iti). Hence we 

 shall work in many cases with the standard form {B). But also the 

 canonical form will often be able to serve us by its greater conciseness. 



We shall now suppose the connection between two particular 

 integrals .i{t) and )/{t) of (.4) to be given in one of the two forms : 



H^'fi-v) (11) 



or 



F{.i',v) = (12) 



The latter equation will serve in particular as basis when FU', y) = 

 is an algebraical equation of order ii. In this case we shall make it 

 homogeneous by introducing the factor of homogeneity c (where after 

 the operations have been performed z is put equal to 1 ) and we 

 shall then write it 



F{,^,;i,z) = () (13) 



Our tirst work is to express the functions J and q^ in one of the 

 two inregrals, e. g. d\ We commence with equation (IT) and we 

 introduce the following abridgments : 



dx dx^ dt^ dt^' dt^ dt^" dt^ 



dx • d^x dy ■ d^y dl ■ 



dx dr^ dx dx^ dx 



The two functions ir{t^) and //(^J satisfy the differential equation 

 which we shall write in the canonical form : 



