136 A. Mukhopadhyay — Differential Equation to all Conies. [No. 2, 



It may not be altogether uninstructive to point out that the ease 

 with which the Mongian is derived above is simply due to the fact 

 that, instead of differentiating (as other writers have done) the equa- 

 tion (1) which is an implicit function of £C and y, we first express 

 y as an explicit function of £C in (3), and then proceed to the differentia- 

 tion. 



It is interesting to investigate the differential equations of all 

 parabolas and circles by the above process. If the curve is a parabola, 

 we have h^ = ab in the general equation (1), and (2) reduces to 



hy = - (Jix-Vf) ± [ ^{hf- hg)x-\-(p - he) ] ^ ; 



which may be written 



y = Pa3-f-Q ±\/Rx-\-S. 



— } , we get 

 _1 R^ ■ 





Therefore, 



so that 



4 (Raj + S)'* 



which is accordingly the general differential equation of all parabolas. 

 When developed, this may be written 





which equation was given by Professor M. Roberts in the Dublin 

 Examination Papers for 1875, (p. 277, Question 3). 



If the curve is a circle, we must have a =■ h, h =. in the general 

 equation (1), so that (2) becomes 



ay = -f± I - a^x^ - 2agx-\-p - 



which may be written, 



y = Q± ^-x^^2Kx-\-B, 

 which leads to 



dy -^+H 



^^' (-x^-\-2BiX-\-B)^ 



d-y^^ -(B + H«) 



dx^ (_,^2^2H^ + B)^" 



