570] 
IN FIVE-DIMENSIONAL SPACE. 
83 
theory of the resultant axis, viz. the rotation round the resultant axis is then 180°, 
and we have 0X=0X', OY=OY', 0Z=0Z', and thence we have evidently YZ'=YZ, 
ZX' = Z'X, XT = X'Y. 
Z 
But to prove it analytically, writing P, Q, R for /3" — y, 7 — a", a' — /3 respectively, 
and O for 1 + a + /3' + 7", observe that we have identically 
(/3" + 7 )0 = QP, 
(7 + a 7 ) O = RP, 
(a' + /3")0 = PQ, 
(/3" + 7')P = (7 + OQ = («' + /3)^ 
(a —1)0 = - 7 Q + /3R , 
a'O =- y'Q +(1+/3')P, 
a"0 = -(l+ 7 ")Q + /3"R , 
/30 = — (1 + a ) R+ 7P , 
(/3'-l)0 = - a'P + 7 'P , 
/3"0 =- a"P + (1+ 7 ")P, 
7O =— /3P + (1 + a ) Q, 
7 O = -(l+/3')P + a'Q , 
(7" — 1) O = — /3"P + a!'Q , 
whence O being = 0, we have also P = 0, Q = 0, R = 0. The final conclusion is that 
the two superlines of opposite kinds, when they intersect, intersect in a line.