1888.] A. Makliopadliyay— D(/femz^m? Equation of all Parabolas. 331 



§ 4. Miscellaneous Theorems. 



The differential expression 



^iis - 5r2, 

 tlie vanisliing of wHcli we find to be the differential equation of all 

 parabolas, may appropriately be taken to represent the species of the 

 conic of closest contact at any point of a given curve. For, from the 

 equation 



ax^ + 21ixy + &?/2 + 2^07 4- 2/V 4- c = 0, 



we have 

 where 



2/ = P^ + Q ±^Aa'2 + 2Ha; + B , 



whence we bave, as usual 



dhi AB - H2 



(Ax2 4- 2Ha3 + B) •' 



3 (AB - H2) (Ao; + H) 

 r = + — , 



(A^2 + 2Ha; + B)^ 

 3 (AB - H2) I 4 (Ax + H)3 - (AB - H2) | 



(Aa;2 4- 2Ha3 + B)^ 

 Therefore, by actual calculation, we get 



K 2 Q 9A(AB-P7gp 



^'' - -^^^ = {A^fiT^TTW ' 



SO that it is clear that the differential expression 



5r2 - 3^5 

 is of the same sign as 



A and h^ — ah. 

 HencC; we bave the theorem that at any point of a curve, the conic of 

 five-pointic-contact is an ellipse, hypcrbohi, or parabola, according as 



is negative, positive, or zero.* 



Since we have proved that the radius of aberrancy is given by the 

 formula 



* See Dublin Examination Papers, 1875, p. 279, Ques. 4, by Prof. M. Poberta. 



