134 A. Mukhopadliyay — Differential Equation to all Conies. [No. 2, 



XI. — On Mongers Differential Equation to all Conies. — By Asutosh 

 MuKHOPADHTAY, M. A., F. R. A. S., F. R. S. E., Communicated hy 

 The Hon'ble Mahendralal Sircar, M. D., C. I. E. 



[I^eceived June 30tli ;— Read July 6th, 1887.] 

 §. 1. Introduction. 



Tile present paper relates to the general differential equation of all 

 conies, which was first published by the French mathematician Gaspard 

 Monge in his memoir " Sur les E'quations differentielles des Courbes 

 du Second Degre," (Corresp. sur I'E'cole Polytech. Paris, 1809-13, 

 t. II, pp. 51-54, and. Bulletin de la Soc. Philom. Paris, 1810, pp. 87- 

 88). The subject seems to have attracted the notice of English mathe- 

 maticians, from the following statement made by Boole in his Differ- 

 ential Equations, pp. 19 — 20 ; 



" Monge has deduced the general differential equation of lines of 

 the second order, expressed by the algebraic equation 

 ax^-\' hxy-\- cy^-^ ex-\-fy = 1. 



It is 



/d^y d^_^^diyd_^d_^ /'^V = 



\dxy dx^ dx^ dx^ dx"" \dx^) 



JButf here our powers of geometrical interpretation fail, and results 

 such as this can scarcely he otherwise useful than as a registry of inte- 

 grahle forms. ^* 



It will be noticed that Boole adds no specific reference ; and as 

 the equation was not found, even after diligent search, as well in the 

 printed works of Monge as in his manuscripts, it was at one time 

 believed that Boole had made a misquotation, till Professor Beman 

 pointed out the source of Boole's statement (Nature, t. XXXIII, pp. 

 681-582). But I remark that the matter could have been settled in 

 no time, by a reference to the Royal Society Catalogue of Scientific 

 Papers, where Monge's memoir is actually mentioned (see Yol. IV, 

 p. 441, tit. Monge, No. 22) .* Lastly, it is to be noted that the subject 

 lias been very recently considered by Professor Sylvester, in his bril- 

 liant Lectures on the Theory of Reciprocants, which have been reported 

 with commendable promptitude by Mr. Hammond in the American 

 Journal of Mathematics (See, in particular. Vol. IX, pp. 18-19). 

 §. 2. Derivation of the Mongian. 

 We shall first consider the question of deriving the Mongian from 

 the equation of the conic ; the known methods are more or less lengthy 



* Monge's Equation was also noticed by Lacroix ; see his great work Traiti 

 du Calcul Biffirentiel et du Calcul Integral, Paris, 1810—1819, t, III, pp. 698—699, 

 as a note to § 634, t. II, pp. 371—372 ; I may add that Lacroix gives the reference 

 to Monge's original memoir. 



