466 
MATHEMATICS: H. S. WHITE 
Now it is necessary to express the coordinates of each plane as three- 
rowed determinants in the coordinates of the three points which it 
contains, to simplify, and compare the left hand product of four deter- 
minants in (3) with that on the right, and to show that equations (2) 
above will suffice to prove the equality so reduced. On the left: 
(1257) = ( (ade), (afg), {cdg), (ahc),) 
^ (adeh) (afgc) (cdga) — (adec) (afgh) (cdga) 
= (cdga) { (adeh) (afgc) - (adec) (afgh) ], 
(3457) ^ (cdgh) { (hdfc) (hega) - (hdfa) (hegc) ], 
(1467) = (adeh) (hegc) (cefa) -f (adec) (hega) (cefh), 
(2367) ^ (afgh) (hdfc) (cefa) + (afgc) (hdfa) (cefh). 
On the right: 
(1267) ^ (cefa) { (adeh) (afgc) - (adec) (afgh) } , 
(3467) ^ (cefh) { (hdfc) (hega) - (hdfa) (hegc) ], 
(2357) ^ (afgh) (hdfc) (cdga) + (afgc) (hdfa) (cdgh), 
(1457) = (adeh) (hegc) (cdga) -h (adec) (hegc) (cdgh). 
Notice that on the left two factors in the { 1 are identical with two 
in { } on the right. Exclude these, when multiplication gives in each 
member four terms. Two terms on the left are identical with terms 
on the right. Remove these, and transpose so as to exhibit the factor 
(cefa) (cdgh) - (cefh) (cdga) 
on each side. Neglect this, and our question is reduced to the following. 
Is it true that 
(cI'Mq) ((^hde) ■ (hcdf) (hceg) • (acdg) (acef) 
= (ahdf) (aheg) • (hcdg) (hcef) • (acde) (acfg) ? (4) 
Two appHcations of relations like (2) prove the truth of this, for ex- 
ample we use first b, then c, as seventh point like the g of (2); 
(ahfg) (ahde) (hcdf) (hceg) = (ahdf) (aheg) (hcfg) (hcde), 
and 
(hcfg) (hcde) (acdg) (acef) = (hcdg) (hcef) (acfg) (acde). 
These two Pascalian relations upon the seven points are part of the hy- 
pothesis, hence the Pascalian equation (3) upon the seven planes is veri- 
fied and the theorem is proved. 
It is interesting to restate this theorem somewhat more fully. The 
converse is true by duality, from seven planes to any seven points that 
lie on all 21 of their lines of intersection. Consequently 
// seven planes intersect three and three in seven points of a twisted cubic, 
each of the 29 other sets of seven points that contain all the 21 lines of inter- 
section of those planes is upon another twisted cubic curve. 
