1883.] A. Mukhopadhyay— Jlfo».(/e's Differential Equation. 181 



VII. — The Geometric Interpretation of Mange's Differential Equation to all 

 Conies. — By AsuTOSH Mukhopadhtat, M. A., F. R. A. S., ¥. R. S. E. 



[Received May 22nd j— Read June 6th, 1888.] 

 Contents.* 

 § 1. Historical iutrodaction. 

 § 2. Geometric interpretation. 



§ 1. Historical Introduction. 

 Before proceeding to give the true geometric interpretation of 

 Monge's differential equation to all conies, wliicli I have recently dis- 

 covered, and which it is the object of this paper to announce and establish, 

 a brief survey of the past history and present position of the problem 

 may not be wholly unprofitable. In the first place, then, we remark 

 that the differential equation of all conies was, more than three-quarters 

 of a century ago, first discovered by the illustrious French mathema- 

 tician Gaspard Monge, and published by him in IBIO.t It should be 

 remembered that, in his paper, Monge does not furnish us with any clue 

 to the method by which, from the integral equation of the conic, he 

 derived the differential equation which now ajopropriatoly bears his 

 na,me : neither is there any attempt at a geometric interpretation ; it 

 is simply stated that the differential equation to all conies of the second 

 order as obtained by the elimination of the constants from the equation 

 Ayi + 2lixy + 0*2 -f SDi/ + 2E.» + 1 = 0 



is 



9qH - 452rs + 40;-3 = 0, 



where, as usual, 



% dSy d*y dh, 



^ ~ dx' ^ ~ dx^' ~ dx^' - rfx*' - ' 

 and this statement is followed by a verification that the differential 

 equation of all circles 



(1 + p^')r = 3i.f 

 leads, on differentiation, to the differential equation of all conies. 



After Monge's paper, we come to the following statement made by 

 the late Dr. Boole :J 



» For a fall analysis of this paper, see the Proceedings for 1888, pp. 157-158 ; 

 Bee also Nature, vol. xxxviii, p. 173. 



t Sm- les Equations differontielles des Oonrbes da Second Degre. (Bulletin de 

 la Soc. Philom. Paris, 1810, pp. 87-88 ; Oorresp. sm- I'E'cole Polytech, (Hachette) 

 Paris, 1809-13, t. ii, pp. 51-54). 



J Differential Equations, Fourth Edition, pp, 19-20. 



