144 On the Differential Equation of a Conic. 



we take the 4th, 5th, 6th, 7th, 8th, and 9th differential coeffi- 



cients, and find 



My + N/3 + Pa + Q* 2 

 M$ + Ny+P/3+Q.2«/3 

 Me+NS+P 7 + Q(2 ar +/3 2 ) 

 M? + Ne + PS + Q(2*S + 2#y) 

 M7 ? + N?+Pe+Q(2«6 + 2 i 8S + y 2 ) 



= 



A 



= 0, 

 =0, 



= 



+ Ra 2 



+ B.2«/3 + S« 3 



+ E(2« 7 + /3 2 ) +S.3a 2 /3 



+ ~R{2*S + 2/3 7 ) + S(3* 2 7 + 3«/3 2 ) 

 M^ + N7 7 + P? + Q(2<+2/36 + 2 r S) + R(2ae + 2 i ea + 7 2 ) + S(3a 2 8 + 6«i3 r + ^ 3 ) = 0;i 



where 



M = bi + c x x + 2c 2 y + d x x 2 + 2d 2 xy + 3% 2 , 



N = ci + 2c 2 ?/i + 2^ + 2d 2 <^i + 2d 2 y + Brf^i, 



P=^ 1 + 2% 1 + 3% 2 , 



Q 1 = c 2 + d 2 x + M s y, 



~R=d 2 + 3d 3 y 1} 



S =«*..; 

 and where consequently, as it is most important to observe 

 for the purpose of deducing any one equation from the pre- 

 ceding, 



dx ~~ ' dx 



f=4R«, <* Q 



dx 7 



Eliminating M, N, P, Q, R, S, we have the required differ- 

 ential equation 



= 2P + 2Q?/2=2P + 4Q«, 



= R, ^ = 6S«. 



dx dx 



7 j3 « « 2 



$ 7 /3 2*0 



e 8 y 2*y + /3 2 



f e S 2«S + 2/3y 



t; ? e 2*e + 2/3S + y 2 



** 

 2c$ 



2*y + j3 2 

 2*S + 2/3y 



3**0 



3« 2 7 + 3a/3 2 



= 0. 



v ? 2«?+2/3e + 2 r S 2ae + 2^S + 7 2 3a 2 S + 6«/5 r + /3 3 

 4. The equation of the conic, written with like symbols, is 



7 « 2 



8 7 2«/3 



In both we note as part of the law of formation of the 

 elements, that when one non-zero element of a column has 



