1888.] A. Mukhopadhyay — Differential Equation of all Parabolas. 329 



cos 8 "" q 



But tlie equilateral hyperbola being a conic, we have from tbe preceding 

 investigation 



whence 



sin 



g _ 3pgg - r (1 -f p^) 



. 3r/ 



cos = 



Therefore we see that the distance of the centre of the osculating equi- 

 lateral hyperbola from the given point (which is the origin) is furnished 



^ ^ - 3^ (1+ p^) 



I ,.2 4- (,p _ 3^2)2 I 2 



Hence, the coordinate axes being the tangent and normal at any 

 point of a given curve, the values of the coordinates of the centre of the 

 osculating equilateral hyperbola at that point are given by 



3q \/l + p2 I r (1 + ^2) _ 3^^^2 

 X = 



Y = 



r% 4- (rp - 3^2)2 

 Spqr >/l + p^ 



r^ -+- (7y — 3g2)2 



If the coordinate axes, instead of being the tangent and normal at the 

 given point, are such that the axis of cc makes the angle 9 with the tan- 

 gent, we have 



dy 



tan ^ = - -^ = - ^ 

 dx 



. /, — P /I 1 



sm 9 — ■ , cos 6^ = 



\/l +p2 V 1 + p^ 



and the new coordinates of the centre of the osculating equilateral hyper- 

 bola are given by the two expressions 



3qr (1 + p^) 

 7-2 + Op - Hh^ 

 Sq (1 + p^) (pr - 3^/) 

 7-2 + (rp -3r/)2 



We, therefore, finally infer that if a curve be referred to rectangular 

 43 



X cos ^ + Y sin ^ = 

 - X sin ^ + Y cos ^ = 



