1889.] A. Miikhopadhyay— Ifougie'* Bifferenlial Equation. 183 

 ■where 



To = (l+i^^) (3^^ _ _ (2pr - Sg^) 



aud, by actual calculation, I have proved this differential equation to be 

 one of the five independent first integrals of the Mongian equation* 

 Professor Sylvester's interpretation is, similarly, wholly out of mark, 

 as satisfying the second test but not the first, inasmuch as it gives a 

 property not of all conies but of an extraneous curve. 



So far my criticism has been purely negative, as I have confined 

 myself to the statement that the true interpretation of the Mongian 

 equation still remains to be found. I now proceed to give what I be- 

 lieve to be the long sought for interpretation of the differential equation 

 to all conica. 



§ 2. Geometric Interpretation. 

 Consider the conic of closest contact at a given point of any curve ; 

 refer the system to rectangular axes through any origin ; then, if x, y 

 be the coordinates of the given point, and a, /3 those of the centre of 

 aberrancy, I have already established the system of equationsf 

 3qr 



-Sqs-hr^ • 



If now be the angle between two consecutive axes of aberrancy, p the 

 radius of curvature, and ds the element of arc, of the " aberrancy curve " 

 (which is the locus of the centre of aberrancy), we have 

 ds'-da'' + fZ^» 

 ds 



Now, from the above expressions for a, (3 we get easily 



* See Nalure, vol. xxxviii, pp. 318-319, wliero Liont.-Col. Cunningham sub- 

 stantially acknowledges tlio correctness of my criticism. 



+ See my paper "On tlio Differential Equation of all Parabolas", J. A. S. B. 

 (1888), vol. Ivii, pt. ii, pp. 310—332 ; cf. p. 3U. 



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