1888.] A. Mukhopadliay — On Mongers Equation. 157 



derived for the radius of aberrancy and the index of aberrancy. In the next 

 place, expressions for the co-ordinates of the centre of aberrancy, when the 

 curve is referred to rectangular axes through any origin, are written down 

 with ease, and, it is pointed out that several interesting results, includ- 

 ing the equation of the axis of aberrancy, are immediate consequences 

 of the formulae obtained. The third section contains the geometric 

 interpretation ; from the formula for the index of aberrancy previously 

 obtained, it is shewn that the true geometric interpretation of the 

 differential equation of all parabolas, is that the index of aherrancij vanishes 

 at every point of every parabola. The fourth and last section contains 

 a discussion of some miscellaneous theorems ; it is pointed out that the 

 differential expression, the vanishing of which is found to be the differ- 

 ential equation of all parabolas, may appropriately be taken to distinguish 

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

 lastly, the differential equation of all parabolas in terms of the radius of 

 curvature and the angle which the normal makes with the principal 

 axis, is obtained from a result incidentally given in the course of the 

 foregoing discussion ; and, by integrating this differential equation, the 

 known form of the intrinsic equation of a parabola is verified. 



The paper will be published in full in the Journal, Part II, for 

 1888. 



5. The Geometrical Interpretation of Mongers Blfferential Equation 

 to all Conies.— By Babd Asutosh Mukhopadhyay, M. A., F. R. A. I?,, 

 F. R. S. E. 



(Abstract.) 



The object of the author in the present paper has been to establish 

 the true geometric interpretation of the Mongian equation, recently 

 discovered by him. The paper is divided into two sections, of which the 

 first contains an historical introduction, in which a rapid survey is taken 

 of the past history and present condition of the problem ; the review 

 begins with an account of Monge's original paper ; Boole's statement 

 that in the case of the Mongian equation oiir powers of geometrical 

 interpretation fail, is next noticed ; and, lastly, the reasons for rejecting 

 the interpretations of Cunningham and Sylvester, are summarised. 



The second section gives the geometrical interpretation of the 

 Mongian equation ; the most general expression for the radius of curva- 

 ture at any point of the aberrancy curve (which is the curve-locus of the 

 centre of aberrancy) of any given curve, is first calculated by means of 

 the formulae given in the author's paper on the Differential Equation 

 of all Parabolas, of which an abstract will be found above. As an 

 immediate consequence of this formula, it is deduced that the true 



