On the Differential Equation of a Conic. 143 



can be traced. Then 



( * 2 <f>(x)f(x)dx = ( a2 <f)(a0)y!,(ad)d(ae) = 2a 



x area 



between the two curves and the lines 6 — ol^ 6 = ol 2 . 



I am, yours &c, 

 Melbourne, December 11, 1885. WlLLlAM SUTHERLAND. 



T 



XX. The Differential Equation of a Conic. 

 By Thomas Muir, LL.D* 



HE differential equation of a conic, 

 9 \da?) dx 5 ^ dx* d# dx* + 4U W/ ~ ' 



•which Boole attributes to Monge, and to which Sylvester in 

 this week's i Nature ' (vol. xxxiii. p. 224) so fully directs 

 attention, can be very readily obtained as follows. 



2. Let the equation of the conic be 



a Q + b x + b x y + c x 2 + c x xy + cgj 1 = ; 



and let us denote the successive differential coefficients of y 

 with respect to x by y\, y%, ,y 3 , .... Then taking the third, 

 fourth, and fifth differential coefficients of both sides of the 

 equation, we have 



(2c 2 y + c x x + bi)y 3 + (2c 2 y l + c 1 )3y 2 = 0, 



(2c 2 y + c x x + 6i)y 4 + 2 yi + c 1 )4y 3 + 2c 2 . Zy\ 



{2c^y + c v x + h)y 5 + (2c 2 y 1 + c^by^ + 2c 2 . 10y 2 y 3 ■ 



whence at once, by elimination, 



=0 ') 



=0, 

 =0;J 



'2 



y± 4y 3 %y\ 



Vh %4 10^23/2 



or 



y 2 (40z/| - ^y 2 y 3 y^ + 9yjy„) = 0, 



as was required. (Cf Jordan's Cours a" Analyse, i. p. 53.) 



3. The method is perfectly general. To be seen at its best, 

 however, it requires that for y 2 , y 3 , y 4 , . . . we write 2! a, 3!/3, 

 4!y, .... Thus, if the equation of any cubic be 



a + b x + b x y + c x 2 + e x xy + c 2 y 2 + d x* -f c^afy + d 2 xif + d 3 ,y 3 = 0, 

 * Oomuiuiiieated by the Author. 



