372 Dr. C. J. Hargreave on Differential 



Consequently p. and tr are respectively functions of 



(f+{i+x)*y*±x, 



or of t x and / 2 suppose. In fact we have 



tn- = sanie function of t 2 . 



The primitive may be placed in the form 



(«-••**)(«-€— ")=0, 



which, with the above values of sr and /jl, probably admits of 

 being rationalized. At all events we learn that the only func- 

 tions which enter in the complete primitive are 



*+(y+(i +«)*)*' 



a circumstance which it would be difficult to infer from the ordi- 

 nary mode of solution. 



The method has now been sufficiently illustrated in reference 

 to equations of the second degree. I reserve the application of 

 the principles of this paper to equations of the third and higher 

 degrees, until I have had further opportunities of studying them. 



Discussion of the Partial Equation. 



17. The partial differential equation on which we depend for 

 the integration of equations of the second degree, 



viz. r + 20s + $4 + Pp + Qq = 0, 



is worthy of independent consideration. 



It will be found to be soluble by Monge's method. His auxi- 

 liary equations are 



®'-»*£+*-* 



dp dy-\-(f) 2 dy dx + (Pp + Qq) dy dec — ; 



the first of which is the original differential equation of the above 

 paper, for this purpose considered as integrable. This equation 

 is resoluble into 



dy=((f) l -±^R)dcc; 



one of which reduces the second equation to 



D!osjjO+(fc-B) g )+]|lbg(fc+U)^=0, 



in which D is used to denote the complete differential of the 

 function subjected to it. Multiplying this by X, an assumed 



