330 A. Makliopadbyay — Differential Equation, of all Parabolas. [No. 4, 



axes drawn througli any origin, tlie coordinates {^, rj) of the centre of 

 the osculating equilateral hyperbola at any given point (x, y) of the 

 curve, are given in the most general form by the system 



3qr (1 + p'^) 



V = V + 



,.2 _^ (^^-j) _ 3^2^)^ 



3q (1 -I- jo2) (pr - 3g2)2 



The equation of the line joining the centre of the osculating equi- 

 lateral hyperbola with the given point on the curve is at once written 

 down in its most general form, viz., x, y being the coordinates of the 

 point and X, Y the current coordinates, we have for the required equa- 

 tion 



'K — x_x — ^__ r 



^ - y " y-y " pr-Sq^ ^ 



which shews that the centre of the osculating equilateral hyperbola is on 

 the axis of aberrancy, as is also geometrically evident. From the above 

 values of $, rj, it can be shown after some reductions that 



where 





_ ^^ - r^ (1 4- p'^) 



r2 + (rp - 3r/)2 i 2 



r (1 4- p'^) {Qq^- pr ) - ^pq'^ 

 fXQ — 



r% -I- (rp - Sq'^y 



To = 9q^ - Qpq^r + (1 + p^) (Sqs - 4r2) , 

 so that Tq = is the differential equation of all equilateral hyperbolas. 



§ 3. Geometric Interpretation. 



It is now extremely easy to give the true geometric interpretation 

 of the differential equation of all parabolas ; for we have shewn above 

 that the index of aberrancy is given by the formula 



3^5 - hr^ 



3q I r2 + (7p - 32^)2 I 



and the differential equation of all parabolas is 



Sqs - 5r2 := 0. 

 Hence, we conclude that the required geometric interpretation is the 

 property that the index of aberrancy vanishes at every point of every para- 

 bola. 



