1888.] A. Mukhopadliyay — Bemarhs on Mongers Equation. 77 



meaning of the differential equation of any curve. The other inter- 

 pretations given in the Colonel's paper are similarly wholly extraneous. 

 To my mind, the matter appears to be simply this, viz., the differential 

 equation of any curve is nothing but the analytical representation of 

 the vanishing of a certain geometrical quantity in connection with that 

 curve, and the geometrical interpretation is exactly the process of dis- 

 covering what this quantity is ; Professor Sylvester's interpretation is 

 irrelevant as not satisfying the first test laid down above, and Col. 

 Cunningham's interpretation, as satisfying neither of the tests, has surely 

 no better claims to our attention. 



But, gentlemen, it is possible to prove not only that the Colonel's 

 interpretation has entirely missed the mark, but also that it is the in- 

 terpretation of a differential equation very different from the Mongiau 

 equation ; and, guided by the wholly erroneous interpretations which 

 Col. Cunningham has given in the case of the straight line and circle, 

 I have been able to discover the differential equation to which in reality 

 belongs the geometrical interpretation given by the Colonel. In fact, as 

 we have already a priori shewn that the Colonel's interpretation is irre- 

 levant, we may further strengthen our position by shewing that the 

 interpretation belongs to a differential equation, which, though wholly 

 distinct from the Mongian equation, stands in a very important relation 

 to it. 



Let us first take the case of the straight line, whose differential 

 equation is interpreted by the Colonel to mean that the direction of a 

 straight line is the same at all parts ; this, as have already remarked, 

 is totally erroneous. But, at the same time, the geometrical property 

 is obviously the interpretation of the equation. 



dy 



which we at once recognize to be the first integral of 



which is the differential equation of all straight lines. 



Similarly, in the case of the circle, the interpretation given by the 

 Colonel, viz. J the curvature of a circle is constant, really belongs to the 

 equation 



3 



iMirlL, 



dx^ 



