J887.] A. Mukhopadliyay — Differential Equation to all Conies. 141 



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 



ax 



we have 



Z 





Multiplying both sides hjz^y and then substituting z = ^ ^, 



9ci = c, and 9(c2' — c^Cg) = c', we get 



f^Vj^r' r-iy-^ - r y\ 



\dx') ^ \dx') \dx^) 



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. 



§. 5. 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 x 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 -|- hy"" -\- 2gx -f 2/i/ -|- c = 0, 



we have Monge's differential equation. If we interchange a?, ?/, we get, 

 corresponding to the integral equation 



(5) a2/^4-2%^ + 6^H2^2/+2/^+c = 0, 

 the differential equation 



But the equation (5) represents a conic, and as all conies are repre- 

 sented by the Mongian, the Mongian corresponds to (5) ; but, as (6) 

 also corresponds to (5), we see that the Mongian and (6) are identical, 

 or mutually transformable, which establishes the theorem in question. 



