1887.] A. Mukhopadhyay — Differential liquation to all Conies. 135 



and tedious ; the easiest way known is tbat of Professor Michael 

 Roberts, who gave the following theorem in the Dublin Examination 

 Papers for 1876 (p. 269, Ques. 6) : 

 '' Prove that 



\dx^) "^ Xdx'^) \dx^) * 



where c, c' are arbitrary constants, is a second integral of the dif- 

 ferential equation of the fifth order which represents a conic section." 



For Professor Wolstenholme's solution of this question, as well 

 as for the method of obtaining the Mongian by twice differentiating the 

 above equation, see Educational Times Reprint, Vol. XXIV, pp. 

 104-106, Question 4821 ; see also. Professor Burnside's Question 

 7104, in Vol. XXXVIII, p. 71. The method which I propose is as 

 follows : 



Let the equation of the conic be written in the standard form 



(1) ax' +2}ixy -\-hy^ -\-2gx^-2.fy + c=0. 



Solving this as a quadratic in y^ we have (c/. Salmon's Conies, p. 72, 

 Ed. 1879) 



(2) hy=^-(kx-{f)± Uh^-ah)x''-^2{hf-hg)x^-{f-lc)\^, 



which may be written, 



(3) 2/=P^-fQ±\/A^'*+2Ha3-fB. 



(^ \ 2 

 — j , we have 



S = ^S^«A.'+2^^ + ^)'j 



_ AB-H^ 



{Ax^ ^2\lx^By 

 Therefore, 



/0\ ~ 3 ^ + (AB - H=^) ~ ^ (Aoj^ +2Ha5 + B). 

 Operating with I -7-), we have 



©'[(S);']=«. 



which is accordingly the general differential equation to all conies ; 

 if we write it in its developed form, after performing the operations 

 indicated, we have 



\dxy dx^ dx^ dx^ dx* \dx^) 



which is exactly the equation of Monge who wrote it in the now familiar 

 form 



92^^"452r5 + 40r5=:0. 



