140 A. Mukhopadhyay — Differential Eqiiation to all Conies. [No. 2, 

 Therefore 



and tI= I AV-f2H'a3+B' 



"whence, by two simple integrations, we easily pass to 



y = Pa;+ Q + \/aS*T2H7+B, 

 which at once leads to 



the general conic-primitive sought. 



§. 4. Second method of integrating the Mongian. 



We shall next proceed to shew how the Mongian equation may be 

 integrated by means of an integrating factor. The equation being 

 written, as before, in the form 



JJL_ 



if we multiply this by the integrating factor z ^ , it may be written, 

 -^d^z -%dzd'z 4^0 -iJ^ ydz\^ 



z 



dx^ dx dx^ 9 \dx) 



By the application of ordinary methods (Boole, pp. 222 — 226. 

 Forsyth, pp. 82 — 85), the left hand member is seen to be a perfect 

 differential, and, integrating, we get 





z 



■*S-.--^(S/- 



which may be written 









sl'-'ll— .. 



whence 





dx ^ 



Integrating again, 





z ^ = c^x^-2c^x\c^^ 



d^y —^ 



whence ^ ~ J^ = {c^x^ -'2gc^x-\-c^) % 



and the solution may be completed as before by two simple integrations. 



