74 A. Mukliopddliyay — Uemarlcs on Mongers Equation. [Feb. 



of Bengal for 1887 there is a Paper on '* Monge's Differential Equation 

 to all Conies " wherein the author quotes the late Dr. Boole's remark 

 that our powers of geometrical interpretation fail for this equation, (and 

 presumably for nearly all except those of straight line and circle). 



I would refer your readers to Vol. XIV of the Quarterly Journal of 

 Pure and Applied Mathematics for 1877, page 226, wherein I gave three 

 general geometric interpretations of differential equations in general. 



These were as follows : — 



1. The curve whose differential equation is of the m*^ order can 

 be drawn so as to satisfy m independent consistent conditions. 



2. If the m^h order differential equation of a curve F be satisfied 

 at any point of another curve /, then these curves have contact at that 

 point of the m^^ order (one degree higher than ordinary). 



3. All fundamental geometric quantities (e. g. lengths, areas, 

 eccentricities &c) connected with a variable curve F possessing m degrees 

 of freedom and osculating in the highest or (m — 1) degree another fixed 

 curve of same species, similarly conditioned^ are constant right round 

 the latter. 



This last condition is the generalized form of those given by Boole 

 for the straight line and circle : the interpretation of the Mongian thus 

 becomes : — " The eccentricity of the osculating conic of a given conic is 

 constant all round the latter." For proofs, see the paper referred to." 



Babu Asutosh Mukhopadhtay made the following remarks in reply : 

 Mr. President and Gentlemen, 



When my paper on Monge's equation was read before the Society, 

 I was not aware of Lt.-Col. Cunningham's paper, and, in fact, I had 

 not the opportunity of examining it till I had learnt the contents of 

 the letter which has just been read to you. With reference to the letter, 

 and the paper to which we are referred therein, I will remark in the 

 first place that they do not touch upon any of the vital points discussed 

 in my paper. You may remember that my paper on Monge's equation 

 was devoted principally to a consideration of four things, viz., the easiest 

 way of forming the Mongian differential equation from the integral 

 equation of the conic, the integration of the Mongian by ordinary 

 methods,* the permanency of form of the equation, and lastly, a cri- 



* I find that in the Messenger of Mathematics, (Vol. XVII, pp. 118 — 145, Decem- 

 ber 1887 to February 1888), there is a paper by Col. Cunningham on the Depression 

 of Differential Equations, the chief object of which seems to be the solution of the 

 Mongian equation in different ways ; I find that my transformation (Journal, A. S. B. 

 Vol. LVI, Part II, p. 138) is reproduced on pp. 141 — 142, of course, without the 

 slightest acknowledgement that it had been given before by me ; that the Colonel 

 was acquainted with my paper at the date of the publication of his article, is now 

 sufficiently obvious, and his reasons for not acknowledging that the transformation 

 in question had been given six months before by me, are best known to him. 



