324 A. Mukhoipiidhy ay— Diferential JEquation of all Paraholas. [No. 4, 



and the new coordinates of the centre of aberrancy are given by the two 

 expressions 



X cos ^ + Y sin = ^^ " ^Z o 

 Sqs — or 2 



- X sin + Y cos e = -3'l(pr-3q^)_ 



3q8 — 5?-2 

 We, therefore, finally infer that if a curve be referred to rectangular 

 axes drawn through any origin, the co-ordinates (a, yS) of the centre of 

 aberrancy at any given point (cc, y) of the curve, are given in the most 

 general form by the system 



Sc/r 



a z=: X — = 



oqs — 5r2 



Sq (pr - 3gg) ^ 

 ^ " ^ Sqs - 5r2 



The equation of the axis of aberrancy, in its most general form, may now 

 be at once written down, viz., x, y being the coordinates of the point on 

 the curve through which the axis of aberrancy passes, and X, Y, the 

 current coordinates, we have for the required equation 

 X — 33 X — a r 



^ " y ^ y " f^ _pr — 3^^ " 

 It may usefully be noted that the values of a, /8 obtained above, 

 lead to some interesting results, viz., we have 



da _ r (9qH - 45 grg + 40rS) 

 ,, H ~~ {Sqs - 5r2)2 ' 



d(B _ (pr - 8g^) (9r/2i - ^hqrs + 40r3) 

 d^ ^ (3qs - 5r2)2 ' 



so that we may put 



where 



so that 



X = 



dx 



r pr — 3q^ 



(Sqs - 5r2)S ' ^ " (3^5 - 5r2j2 ' 

 T = 9qH - 4ibqrs + 40rS, 



T = 

 is Monge's differential equation to all conies. f It is clear from these 

 two expressions that if the given curve is a conic, we have 



* Cf. Dublin Examination Papers, 1876, p. 152, Ques. 6, bj Prof. M. Roberts. 

 t Cf. Dublin Examination Papers, 1880, p. 361, Ques. 5, by Prof. M. Roberts. 



