320 A. Mnkliopadhyay — Differential Equation of all Parabolas. [No. 4, 



dp K"^(d i q^/^/V-fi-^/^^vi ^M 



dx "" A%\2 I (^o; \da;'^) L \^^/ J ^''^^ V 



Hence, we get 



tan 8 = 5 — = — — 

 op op do) 



dp 

 1 d!a? 

 3p d<j} 



dx 



1 4- 



/dyY ] d^ 

 \dx) ] dx^ 



\dxy 



Using ^, q, r to denote tlie first, second and third differential co-efficients 

 of y with respect to ^, we have the formula for the angle of aberrancy in 

 the no TV familiar form 



^ - (1 + V ^) r 



It is easy to verify this formula when the equation of the conic is 

 given in form 



for the coordinates of any point being a cos <p, h sin <p, the equation of 

 the central radius vector is ; 



ay cos 'p^bx sin <^, 

 and the normal is 



cos <p sm <p 



so that the angle between these two lines is given by 



a^—b^ 



tan 8 = = — sin <p cos </>. 



ab 



Again, from the equation of the curve we have 



pz= — y = cot <p. 



^ a ^a^ - ^2 



2=- 



a 

 ab b 



(a* — x*Y 



3a6aj __ 3?) cos ^ 



