146 Proceedings of the Ro>/al Irish Acadeinij. 



Imagine the cubic expressed in the form 



4 



^ (/,-« + niry + nrzf, 



and that xi^'^V^) "^ ^ ^^ ^^^® equation of a conic which passes through the 

 four points (///?ii7i-,) . . . (^im/rti), then \ ("'1/-) is an annihilating conic, for 

 the result of operating with 



d d d 

 X 



L 



h 



h 



nu 



m-i 



mi 



012 



n^ 



Ui 



dx dy dz 



is the sum of four terms like {li-c + mj?/ + Uiz) xihrnini), which is zero. 



If an annihilating conic x pass through one of the points (Um^n^), &c., it 

 passes through each of the points [hm^n,), ko,., for since the conic yields an 

 annihilating operator we have 



%^ [IrCC + TJlry + 7?,.,?) ^ (/,• //?;•«,■) = 0, 



whence the equations 



where ^i is zero, and therefore ^2 = ^*' Xs = ^> Xi = ^> ^^i^less 



= 0. 



This latter equation would imply that the hessian of the cubic had a double 

 point ; for it is easily seen that every pair of the tetrad of lines hx + niiy + n^z, 

 &c., meets on the hessian. 



If then U, V, IV be three annihilating conies, we may take 



to be two conies which pass through four points whose coordinates are 

 possible values for Umiih, &c., thus giving a reduction of the cubic to the 

 sum of four cubes. 



Any point oi'yz in the plane is in general a possible position for a pole 

 [limini); for suppose the cubic u to be = Sj/ (avc' + y/ + s'/)', then, as 

 u - p' [xx + yy' + zz'y is the sum of three cubes, its invariant ^S" vanishes; 

 writing out this invariant we have an equation to determine j/, we find 

 that the coefficients of y^^/^/^ vanish, and obtain the result i/P' = >S', 

 where P' is the result of replacing the tangential coordinates by xy'z in the 

 equation of tlie Cayleyan. 



