1888.] A. Mukhopadhyay — Differential Equation of all Parabolas. 325 



T = 0, 



whicli shews that a and )8 are both independent of x, as is, indeed, 

 geometrically evident, since the osculating conic of a given conic being 

 the curve itself, the centre of aberrancy is a fixed point, viz., the centre 

 of the given conic. Similarly, if 



A, = 00 , /ot = CO , 

 we must have 



Zeis - hr^ = , 

 which shews that the given curve is a parabola, and, then the centre 

 of aberrancy has its coordinates infinite, viz., the centre of aberrancy 

 is the centre of the parabola which is, of course, at infinity. We may 

 also easily find the values of 



da d^ 



dy ' dy' 



viz. J we have 









da __ 



da dx 1 da ^ 





dy "~ 



dx dy ~~ p dx ^ * 





df3 

 dy - 



d^ dx ^l^ d_^ ^ ^^ 

 dec dy p dx 



where 









\ 



X r 





" p " p {3qs - 5r3)2 ' 





/^i 



ju, pr — Scj^ 





" p ~ p CSqs - 5r''f ' 



and, these results shew that when, as before, 



T = 0, 

 the centre of aberrancy is independent of y, and, when 



Ai = CX) , /Xj = GO , 



it is at infinity. 



The directions of the principal axes of the osculating conic are also 

 easily determined, for the conic being 



ax^ + 2hxy + by^ -[■ 2gx -^ 2fy -^ c = 0, 

 if be the angle of inclination of the axis major to the axis of x, we have 



tan 29 = -^ . 

 a — 



But, I have elsewhere* calculated the values of the constants on the 



right hand side in terms of the differential co- efficients, viz., we have 



^^ _ " _ 2 ^1 



where 



* P. A. S. B. 1888, pp. 82—83. 



