126 Journal of the Mitchell Society. [Dec. 



The first step is to find the polar planes of a point (x„y„ z t , w x ) 

 with respect to each one the two triads abc, fgh, then to 

 identify the equations of these two planes, in order to deter- 

 mine the position of the point 0. 



First consider the triad abc, given by the equation 



w (256 * — 384 y — 96 z + 459 w) 



(96 x — 64 j - 256 z + 153 w) = 



which becomes, on clearing out, 



F (#, y, z, w) = 24576 (x 2 + f + z 2 ) w + 70227 w 3 - 53248 #ya/ 

 —74752 xzw + 83232 A-a> 2 -f 104448 yzw 

 -88128 jy a/ 2 - 132192*^ = 



The polar plane 



8F , 8F , SF , 8F 



of the point (a- x , j x , z x , «/ x ) with respect to this system of three 

 planes is 



(49152 x x w x - 53248 ^ x w x - 74752 *,*>, + 83232 w 2 ) x 

 + (49152 ^w, - 53248 a? x w x -f 104448 * x w x — 88128 w/)^ 

 + (49152 ^w, - 74752 at, w x + 104448 j x w x - 132192 w/) * 

 + [24576 (x? + y t s + ^ x 2 ) + 210681 o/ x 2 - 53248 x z y t 



- 747S2x i z i + 166464 at, w, -f 104448 /,*, — 176256^ w, 



- 264384 ^a/J w = [1] 



Also the polar plane 



8/8/ 8/ 8/ 



6X 1 6yf 6Z Z 6W J 



of the point (a* x , _y x . ^ x , w x ) with respect to the system fgh, 

 given by the equation 



(32 x -f 27 -a/) (64 y — 51 w) (32 z - 17 w) = 



or /(at, y, #, «/) = 65536 Ary# -f 55296 _y,ew — 52224 xzw 

 — 44064 ,ew 2 — 34816 #yw — 29276 yw 9 

 + 27744 a-w 2 + 23409 w 3 = 



