141 
1887.] A. Mukliopadhyay —Differential Equation to all Conics. 
It is worth noting that though this second method is apparently 
much shorter than the first method, it may seem to be rather artificial 
in the absence of any clue to the discovery of the proper integrating 
factor; the process, however, has the merit of furnishing an immediate 
proof of Professor Roberts’ theorem, quoted above in §. 2. Thus, since 
_ — Z 
z 3 — = -3c 1 x + ‘3c 2 , 
dx 
we have 
, r] % \ 2 
/— \ = 9c 1 2 a3 2 — 18c 1 c 2 ^ + 9c 2 8 
= 9o 1 (c 1 a3 3 — 2c 2 x-\-c 3 ) + 9(c 8 2 — c 1 c 3 ) 
_ 2 . 
= 9 C x z 3 +9(c 3 2 — CqCg) . 
1 o 
Multiplying both sides by z 3 , and then substituting z 
d *y 
dx** 
9 c 1 = c, and 9(c 2 9 — c 1 c 3 ) = c, we get 
which is exactly Roberts’ theorem quoted above ; and this not only 
shews that the Mongian can be derived from this equation, but also 
that it is a second integral of the Mongian. 
Permanency of Form. 
Professor Sylvester has remarked that the Mongian equation has 
permanency of form, that is to say, if we seek the transformation of 
the Mongian when y is the independent and x the dependent variable, 
the required formula is obtained by interchanging # and y in the 
Mongian ; this theorem, which is proved from the properties of pro¬ 
jective reciprocants, may easily be established as follows. Correspond¬ 
ing to the integral equation 
(4) ax* -4 %hxy -f by 1 + 2gx + 2/y + c = 0, 
we have Monge’s differential equation. If we interchange x, y , we get, 
corresponding to the integral equation 
(5) ay 2 + 2hyx bx*-\-2gy-\- 2fx -f c — 0, 
the differential equation 
( 6 ) 
9 (£)' 
d 5 x 
- -45 _ __ 
dy 5 dy a dy 
d 2 x d s x d*x /d 3 x 
dT + W 3 
) = 
= 0 . 
But the equation (5) represents a conic, and as all conics are repre¬ 
sented by the Mongian, the Mongian corresponds to (5); but, as (tf) 
also corresponds to (5), we see that the Mongian and (6) are identical, 
or mutually transformable, which establishes the theorem in question. 
