Johnston — Method of Obtaining the Cubic Curve. 67 

 we must have 



dx dy dz) \ dx dy dzj 



d X, <^ >,, <^\ f d d ,, d\ 



dx dy dzj \ dx dy dz ' 



d_ , d_ „ ^\ ^ A _^ w ^ w, ^\ 

 dx dy dzj ~ \ dx dy dzj 



Equating the coefficients of x, y, z in these identities we get nine 

 equations of the form 



va' + v'h' + v"g' = ixa" + fjlh" + ix"g". 



The eliminant of these equations with respect to X, /x,, v, X', &c., 

 is a skew symmetrical determinant of the ninth order, which con- 

 sequently vanishes indentically. Therefore the nine equations are 

 not independent, but are connected by a linear relation. It is therefore 

 possible, in general, to transform the conies to the required form, the 

 transformation being given uniquely by any eight of the equations. 

 If the values obtained for X, jx, &c., are such that 



X jU, 1/ 



X' [X.' v' 



X" fx" v" 



the transformation fails, and it is not possible to obtain a cubic having 

 the three conies as polar conies, for the vanishing of this determinant 

 would imply that a; = 0, y = 0, s = 0, passed through a common point. 

 Let 4> be what <^ becomes when we replace X, V, Z in it by the 

 corresponding values of x, y, z. Then since 



we have 

 but 



$ = XV'+ YV+ZTF, 

 <f) ^ Xu + Yv + Zw, 

 x = XX+ fxT+vZ, 

 y = k'X + ix'Y+v'Z, 

 z = \"X + fx"Y+v"Z. 



K.I.A. PKOC, VOL. XXIV., SEC. A.] 



