427] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 473
which expanded, becomes
(a'au' + ßß3 + 77» + SS») (/3/30 + a'a0' - - 7y0)
— (a'3o — ß<h ~ y^o + 87») (/3a0' - a'ß0 + 8y„ - y50)
= (a'a0' + ßßtf - (v7o + S80)2 + (a'ß> - a0'/3)2 - (8y0 - 7g»)2
= a'2a0'2 + ß2ß/ + a'2 ß,? + a„'2ß2 — y2yy — S2802 — — 72S02
= a'2 (a0'2 + /302) + ß2 (a0'2 + /3„2) - y2 (y2 + 802) - (y2 + M
= (a'2 4 /32) (a0'2 + Ä2) - (?2 + 52) (%2 + B2);
but, a'3+/S2+72+S2 = 0 ;
whence this expression is
(a/2+ß2) («o'2 + Ä8+Vo2+^o2)=O.
On the supposition that the vectors were homonymous, i.e. both right or
both left, the corresponding determinant would have been
a' — ß — y —8
ß a - 8 y
a»' ~ ßo - 7o ~ $>
ßo “o' - 70
Squaring, we get, as before,
[a'a0'] [/3a0']
[&«'] [ßßo]
but now,
[“'«o'] = [M]>
[Äa] = - [ßa0'J;
whence the determinant reduces to
[a'a0']2 + [/Sa»']2,
a value very different from that in the former case.
427. The Composition of Vectors.
Let an object x be conveyed to y by the operation of a vector, and let the
object y be then conveyed to z by the operation of a second vector, which we
shall first suppose to be homonymous (i.e. both right or both left) with
the preceding. Then we have, from the first, supposed right
+ ßx.2 + yx3 + S«4,
ya = — ßXi + ax2 + bx3 - yxit
ys — — yXi — 8x2 + a«3 + ßx4,
yt — - 5«! + yx2 — ßx3 + a«4)