Equations of the First Order, 361 



must be a function of x and y. Therefore 



1 + ^l^ = + Hztyj 



or 



dy 9x dy ~ dy' 

 dfi dfM _ ^d™ 



As these equations must be symmetrical with regard to fi and 

 OT, as are the equations from which they are derived, R must 

 have a positive sign in one of them, and a negative sign in 

 the other; and the results may be expressed in either of the two 

 equivalent systems, 



dfi _ 1 (dtn dvr\ 



dfi _ 1 /j, d 

 and 



1 / , dvr dix\ 



div __ 1 /dfi , dfi\ 

 d^~U\dx +(p ^)> 



the intrinsic sign of R being immaterial. 



6. These are the equations which for all quadratic differential 

 forms exhibit the connexion which I endeavoured to explain in 

 the introductory observations. They constitute a species of 

 condition which, we must satisfy before we can return to the 

 complete primitive in a rational and connected form. 



Considered in the light of a condition, the two equations of 

 the system are really one. For remembering that c is fi + vT (or 

 a function of it), we have from them 



dc _dvr f ,-rn^ 



whence 



dc . dc , , t> 



which is the original differential equation in its resolved form. 



7. This connecting system lies at the root of the theory; and 

 I may be excused for giving another proof of its existence. 



