1888.] A. Mukhopadhyay — Differential Equation of all Parabolas. 317 



seems not wholly unnecessary to point out that what we are required to 

 do is simply the discovery of a property of the parabola, leading to a 

 geometrical quantity which, while adequately represented by the above 

 differential expression, vanishes at every point of every parabola. As 

 the interpretation I propose to give, follows directly from the properties 

 of the osculating conic of any curve, I will begin with a brief account 

 of Transon's Theory of Aberrancy as expounded in his original memoir.* 



§ 2. Transon's Theory of Aberrancy. 



Consider the conic of closest contact at any point P of a given 

 curve ; if NP be the normal to the conic at P, and its centre, the line 

 OP is called the axis of aberrancy, the point O the centre of aberrancy, 

 and the angle NOP the angle of aberrancy, viz, this is the angle which 

 measures the deviation of the curve from the circular form. Again, 

 from the closely analogous case of the circle of curvature, we may 

 borrow a very useful term and call the length OP, which joins P with 

 the centre of aberrancy, the radius of aberrancy ; and the reciprocal of 

 this radius may conveniently be termed the index of aberrancy. -^ Simi- 

 larly, the locus of the centre of aberrancy as P travels along the given 

 curve, may not be inappropriately termed the aberrancy curve. Before 

 proceeding to obtain analytical expressions for these geometrical quanti- 

 ties in connection with the osculating conic, we shall first prove the 

 following lemma : 



If 8 be the angle between the central diameter and the normal at 

 any point of a conic, p the radius of curvature, p' the radius of curvature 

 at the corresponding point of the evolute, we have 



tan = - - . 

 3p 



Let C be the centre of the conic, and P the given point on the 



perimeter ; jp the perpendicular from the centre on the tangent at P ; r 



the central radius vector CP ; n the normal PN" as limited by the axis 



major ; w the angle which the normal PN makes with the axis major, 



and 8 the angle CPN. Then, we have the well-known relations 



p z=L r cos 8 



jp2 — q9. cos'** o) -|- 62 sin2 CO = a^ (1 — e^ sin^^ o>) 



* Recherches sur la Courhure des Lignes et des Surfaces, Journal de Mathematiques, 

 {Liouville) ler Ser., t. VI (1841), pp. 191-208. For a very short notice of the subject 

 by Prof. Cayley, see Salmon's Higher Plane Curves, p. 368 (Ed. 1879). 



t In the case of the circle of curvature, the very expressive phrase " index of 

 curvature," which is the reciprocal of the radius of curvature, has been now abridged 

 into the single short term " curvature j" but whether anything has been gained by 

 the change is doubtful, 



