318 A. Mukhopadhyay — Differential Equation of all Paraholas. [No. 4, 



63 1 



n = -^ 



^ \/l - e^ sin^ io 



Hence 



and 



whence 



p a s/ 1 — 6^ s 



cos 8 "" cos 8 



sin'<* o) 

 r = " 



sin (o) — 8) w &^ cos 



sin 0) r a^ 1 — e^ sin^ w * 



e-* sm o) cos w 

 tan = 



Now, it is well-known that the element of arc of the ellipse is given by 



a5 = ^ , 



« n - 



whence 



(l-e'sin^o))^ 

 _^_5» 1 



which give 

 Hence, finally, 



di^ a' (l-e2sin3(o)^' 

 ^ c^p 3Z>2 e^ sin w cos w 

 "" cZ<o"" a (i_e» sin'^o))*' 



p' 3e^ sin w cos co 

 p "~ 1 — e'^ sin" (o 



tan 8 = - — , 

 3p 



and thus the formula is seen to be true for a central conic. To establish 



the property for a parabola, we notice that the centre being now at infinit}^, 



the angle at any point P between the normal and the central radius 



vector is the angle between the normal and the diameter, which is equal 



to the angle which the normal makes with the principal axis ; hence, 



we have 



8 = w. 



Bat the intrinsic equation of the parabola is well-known to be given by 



ds 2a 



do) cos^ow ' 



where 4a is the latus-rectum. Hence, 



2a 



cos^w 



dp 6a sin w 





doj cos* w 



so that 



