1888.] A. Mukhopadhyay — Differential Equation of all Paraholas. 321 



wliich give 



r a sin <p cos ^ 



%2 ~ ~ ~ b ' 



^ a2sin2(^ 



(1+^9^)7* a^ - 



so that 



3^2 ab 



'Sq 



sin <^ cos <l> 



(1+i^ 



whicli is the formula to be verified. 



j^ext, to calculate the radius of aberrancy R, let doi the angle 

 between two consecutive normals, and d\\j the angle between two conse- 

 cutive axes of aberrancy ; then, we have clearly 



do) =z d^ + dS. 

 Again, consider the triangle formed by two consecutive radii of aber- 

 rancy and the element of arc of the given curve ; then, we have 



n _ ds_ 



sing -a) 



And, similarly, from the triangle formed by two consecutive normals and 

 the element of arc of the given curve, we get 



ds = pdoi, 

 whence 



But from the equation 



doi 

 ±1 = p cos o . — — , 

 d^ 



we have 





tan 8 



_ I dp 

 3p doi^ 











sec2 8 



dS _ 1 

 'do) 3' 



cPp 



\d0)) 



or substituting 



for 8, 



we get 



dS _ 

 do) 



d^p 

 9p^^ + 



m 



• 



Hence 





dm 



dm 







42 



