140 A. Mukhopadliyay —Differential Equation to all Gonics. [No. 2, 
Therefore 
and 
whence, by two simple integrations, we easily pass to 
y = P*+ Q ±\/ + 
which at once leads to 
ax* -\-2hxy -\-by 2 -\-2gx-\-2fy c = 0 , 
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 
AK dzd*z . A / dz\ a 
9 * *r= _45 ^^ +40 (&) =0, 
_JLJL_ 
if we multiply this by the integrating factor 0 3 , it may be written, 
-4 d*z 
\ 3 
\ = 0 . 
,40 -V /^ 3 
dx* Z dx dx 2 9 \cLx/ 
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 
— 4 d*z 
z “ - 7-5 — \z 
(mQ(j 
— 4 jdz 
(I)=- 
3c 1? 
which may be written 
d_ 
dx 
-4 dz 
z 
dx 
-4 dz 
—- = —3c lt 
whence 
~— — 3c, x 1 3c?> 
dx 
Integrating again, 
whence 
z — C-^X 2c gC C £) 
= 77 = — 2 c a a>+c s ) a , 
dx 
and the solution may he completed as before by two simple integrations. 
