Equations of the First Order, 363 



8. When the original differential equation is resolved into 

 />=(&±R), 



or 



dy=(cj) 1 +'R)dx, 



dc , . 

 ors are — derivt 

 dy 



derived from c = /jl— ts ; and these are 



dc 



the two integrating factors are — derived from c=yit + 'GT, and 



dc 



dy 



or similar functions of fi. Integrating factors having this rela- 

 tion may be called the connected integrating factors. 



9. If we take either of the two equivalent fundamental 

 systems, and differentiate the first equation with regard to x, 

 and the second with regard to y, and equate the two results, we 

 obtain the linear partial differential equation of the second order, 



d 2 z d 2 z d^z v dz >r\dz__~ 



dx 2 ^ dx dy ^ 2 dy 2 dx dy~~ • 



in which we have written 

 Pfor 



and 



Qfor 



\ dx dy / 



\ dx dy /' 



As this partial equation is satisfied by z — ts, or by z=fA, the 

 complete solution of p 2 — 2(/> 1 ^ + (/> 2 =0 is thus seen to depend 

 on the discovery of any solution whatever of this partial equation. 

 If we can find any function of x and y, or either of them, which 

 proves to be a particular solution of this partial equation, and 

 call it <bt, we then obtain p from the fundamental system ; and 

 it is of course also a solution of the partial. Our object in the 

 particular case is then accomplished. 



Conversely (to digress for a moment from the main subject), 

 if the complete solution of 



is known in the form 



then the general solution of the partial equation must be 



