1888.] A. Mnkliopadliyay — Differential EgfuaHon of all Parabolas. 32S 



Therefore, finally, we get 



9^2 I ^.2 4. (rp - 3^2)2 1 



R3 - ! L . 



Hence, it is evident that if I be the index of aberrancy, that is to say, 

 the reciprocal of the radius of aberrancy, we have 

 ^ Sqs - 5r2 



Sq { 



+ {rp — Sq^/^ 



It is hardly necessary to point out that, as these forniulaG hold when the 

 origin is anywhere, they are true when the origin is taken to be tho 

 given point on the curve whose osculating conic we are considering. 



If we take the tangent and normal at the given point as the axes 

 of X and y respectively, we may easily obtain expressions for the coor- 

 dinates of the centre of aberrancy, viz., we have 



X = R sin 8, y = R cos 8, 

 and from the relation 



(I + p ^) r 

 tan6=p S^T- ' 



we get 



. ^ Spq^ - r (1 + p^) 

 sm 8 — , r 



\/l -\r p'^ I r2 -f (rp — Sg;^)^ 



3^2 



COS 8 = — ziizzr 



v/1 + p^ I r2 4- {rp - 3q^y 



Hence, the coordinate axes being the tangent and normal at any point 

 of a given curve, the values of the coordinates of the centre of aberrancy 

 at that point are given by 



3q iSpq^ -r (I + p^) 



X = 

 Y = 



\/l + p2 (^Sqs - 5r2) 

 9r/ 



\/i +/' (Sqs-^r^) 

 If the coordinate axes, instead of being the tangent and normal at 

 the given point, are such that the axis of x makes an angle with the 

 tangent, we have 



tan ^ = - -p = - p , 

 ax 



-p 1 



sin 6 — — - , cos — — . j 



