326 A. Mukliopadhyay — Differential Equation vf all Parabolas. [No. 4, 



^ence, substiti 



^1 



U 



To 

 itiiig, ^ 



U V 



- s ' ^J3 - To ' 



W 



^3 - Y ' 



=z Sqs - 5r2 , V = 3^25 - 

 W =z Sqh' - pY. 

 = 9q'^ - Qpq^r + (p^ + 

 re get 



- 4r2 , 

 1)V 





tan ^u — 



^% ^3 — Cj — C 



2 V W 



2^ 





2(3.2^r-pV) 



* 





9^* - ^p(^r + (p2 - 1) 

 2W 



V 





- To - 2V • 





The lengths of the axes of the conic of closest contact may also be 

 easily calculated, viz.., the conic being 



ax^ + llixy + by^ + Icjx + 2/y/ + c = 0, 

 and o- the length of either axis, we have the well-known equation 

 ^ A(a + 5) A^ _ 



where A is the discriminant. Now I have alreadyt shewn that 



(7^2 _ ohY 











3 









.re, 



we have 















A (a 

 (/.8 - 



-a6)2 



a + 



6 



= 



i* 



1 





c,^ (7i2- 



-a6)* 



-> (S - 



9* 







H'- 



'^2 



Co' 



(1 









" cj 



7^- 









_ 99^ T, ^ 











Similarly 



* Cf. Dublin Examination Papers, 1876, p. 152, Ques. 5, by Prof. M. Eoberts. 

 t P. A. S. B. 1888, p. 80. % P. A. S. B. 1888, p. 83. 



