182 A. Mukbopadliyay— Vlfferenlud Equation. [No. 2, 



" Monge lias deduced the general differential equation of linos of 

 tlie second order, expressed by the algebraic equation 

 ax^ + bxij + cy^ + ex -\- fy = 1. 



It is 



But, here our poivers of geometrical interpretation fail, and results sucli as 

 this can scarcely be otherwise useful tlian as a registry of integrahle forms." 



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

 maticians from the above statement of Boole, and, during the thirty 

 years which have elapsed since these remarks were first made, there 

 appear to have been two attempts to interpret geometrically Monge's 

 differential equation to all conies. The first of these propositions, by 

 Lieut.-Ool. Cunningham, is that the eccentricity of the osculating conic 

 of a given conic is constant all round the latter.* The second proposi- 

 tion, by Prof. Sylvester, is that the differential equation of a conic is 

 satisfied at the sextactic jpoints of any given curve. f I have elsewhere 

 considered in detail both those propositions, and I have fully set forth 

 my reasons for holding that neither of them is the true geometric inter- 

 pretation of Monge's defferential equation to all conies. J In fact, as I 

 have already remarked, there are two tests which may be apislicd if wo 

 vs^ish to examine whether a proposed interpretation of a given differ- 

 ential equation is relevant or not, viz., 



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

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

 metrical quantity which vanishes at every point of every curve of the 

 system. 



2nd. The geometrical quantity must be adequately represented by 

 the differential equation to be interpreted. 



Lieut. -Col. Cunningham's interpretation cannot be accepted as it 

 satisfies neither of the tests ; it fails to give such a property of all conies 

 as would lead to a geometrical quantity which vanishes at every point 

 of every conic ; moreover, it is not adequately represented by the differen- 

 tial equation to be interpreted, inasmuch as it is really the geometric in- 

 terpretation of the differential equation 



(ei>-2) a_ Tq 

 1-ea ~VU' 



* Quarterly Journal of Mathematics (1877), vol. xiv, pp. 226-229. 

 t American Journal of Mathematics (1886), vol. ix, pp. 18-10. 

 t Journal, A. S. B. (1887), vol. Ivi, part ii, pp. 134-115; P. A. S. B. (1887), 

 pp. 185-186 ; P. A. S. B. (1888), pp. 7-1-80. 



