164 Physical Properties [OH. vn 



shall be a complete differential, and is therefore 



d_ fL\ _ d (M 

 ~ 



from which N can always be found. 



If N is an integrating factor of (395), so that 



where d% is a perfect differential, then we have also 



so that if N is an integrating factor, so also is Nf() in fact N multiplied 

 by any function of is an integrating factor. There are, therefore, an infinite 

 number of integrating factors, 



189. We can, moreover, shew that if N is one integrating factor of (395), 

 so that 



dz = 



then all the integrating factors are included under the form Nf(%). 

 For let N and JV ' be two separate integrating factors, so that 



__ _ 



N~dx' N~dy' 

 and also 



L^^d^ Mi_W 



N'~dx' N'~dy' 

 Then obviously 



r ag 



dx ' "by 

 from which it follows that % is a function of . Hence 



a function of , which proves the result in question. 



190. Applying this to the problem of thermodynamics before us, it 



ears that since -^ and 

 J. 



be a relation of the form 



appears that since -^ and ^r are both of them integrating factors, there must 



=/(&) ................................. (396). 



