76 A. Mukliopadhyciy — Beniarhs on Mongers Equation. [Feb. 



wliere F is a certain function of the variables and tlie differential co- 

 efficients; and, the process of geometrical interpretation of the dif- 

 ferential equation is simply the process of discovering the geometrical 

 meaning of the quantity which we have denoted by F ; in other words, 

 we are required to find out a geometrical quantity, represented by F, 

 which vanishes at every point of every curve of the system whose 

 differential equation is 



F = 0. 

 It is clear, therefore, that there are two tests which may be applied if we 

 wish to examine whether a proposed interpretation of a given differ- 

 ential equation is relevant or not, viz., 



1. The interpretation must give a property of the curve whose 

 differential equation we are interpreting ; in fact, it must give a geo- 

 metrical quantity which vanishes at every point of every curve of 

 the system. 



2. The geometrical quantity must be adequately represented by 

 the differential equation to be interpreted. 



To illustrate these propositions, let us first take the simple case of 

 a straight line ; the integral equation being 



y = mx -\- h, 

 the differential equation is 



and the interpretation clearly is that the curvature vanishes at every 

 point of every straight line. 



Again, in the case of the circle, the integral equation being 

 aj^ -f y'^ -f 2gx ^- 2fy + c = 0, 

 the differential equation is 



and the only true geometrical interpretation of this equation is that the 

 angle of aberrancy vanishes at every point of every circle. 



Let us now take Col. Cunningham's interpretation, viz., the ecceU' 

 tricity of the osculating conic of a given conic is constant all round 

 the latter. From what I have already explained to you, it is clear that 

 this cannot be the geometric interpretation of the Mongian equation ; 

 it fails to furnish a geometrical quantity which, while adequately re- 

 presented by the differential equation, vanishes at every point of every 

 conic ; in fact, it satisfies neither of the fundamental tests I have laid 

 down. I may also point out that the general theorem which Col. Cun- 

 ningham lays down, viz., the constancy of all fundamental projierties of 

 the osculating curve, is, for similar reasons, not at all the geometric 



