1888.] A. Mukhopadhyay — BemarJcs on Mongers Equation, 75 



ticism of Professor Sylvester's geometrical interpretation of the equation. 

 Under this last head, which 1 consider to be the most important part of 

 my paper, I pointed out that Professor Sylvester's interpretation was not 

 anything like the one which had been sought for by mathematicians, 

 and I took care to explain as fully and as clearly as I could, my reasons 

 for differing from that eminent authority. To none of these points do 

 the Colonel's remarks refer.* On the other hand, he takes objection to 

 the statement which I incidentally made that as Professor Sylvester's 

 interpretation cannot be accepted, the true interpretation has yet to be 

 found; and the Colonel claims to have given the true interpretation in 

 a paper published by him eleven years ago in the Quarterly Journal of 

 Mathematics (Vol. XIV, 226 — 229). Before enquiring into the correct- 

 ness or otherwise of the interpretation given in that paper, I may point 

 out that Professor Sylvester's interpretation, which in my former paper 

 was proved to be untenable, was given in 1886, and, while giving his 

 own interpretation, the Professor not only made no mention of the 

 Colonel's paper, but in fact seemed to hold, at least implicitly,t that he 

 was himself the first person to give the true interpretation. Now, this 

 could arise only in one of two ways, viz., either Professor Sylvester 

 had some doubts as to the soandness of the interpretation given by Col. 

 Cunningham, or he was not at all aware of the Colonel's interpretation ; 

 as to the improbability of the latter assumption, I will simply say that 

 Professor Sylvester's name appears as that of one of the editors on the 

 title-page of that very volume of the Quarterly Journal which contains 

 the Colonel's paper. 



I shall now proceed to consider the Colonel's interpretation, and, I 

 may tell you at once that after a very careful consideration of the sub- 

 ject, I have come to the conclusion that it is not at all the true inter- 

 pretation of the Mongian equation. As there seems to be a total mis- 

 conception about the true nature of the process of geometrical inter- 

 pretation of differential equations, I shall first point out as clearly as 

 I can, what I consider to be the only logical and correct view of the 

 subject. In the first place, then, the integral equation of every curve 

 contains a certain number of available arbitrary constants, by assigning 

 particular values to which we may obtain all the curves of the family ; 

 the differential equation, on the other hand, being free from constants, 

 denotes all the curves of the system. Now, it is well-known that the 

 differential equation always comes out in the form 



P = 0, 



* I may mention here that Professor Cayley in a letter to me from Cambridge 

 (Htli September, 1887} remarks about my criticism of Trofcssor Sylvester, that "it 

 is, of course, all perfectly right." 



t See American Journal of Mathematics, Vol. IX, pp. 18—19, 



