468
THE THEORY OF SCREWS.
[422-
and that joining
Let Kj, a2, a3, a4 and ß4, ß2, ß3, ß4 be the co-ordinates of the corners at p
and p". If we substitute ax + XÄ, «2 + Xß2... &c., for x2, x2 &c. in 11 = 0,
2X (aj/Sj + a2ß2 + asßs + üißi) = 0.
Let us make the same substitution in U, we have, in general,
= (11)«! + (12)a;3 + (13)«3 + (14)a;4
= (11) (a, + Xßß + (12) (a2 + Xß2) + (13) (a3 + X&) + (14) (a4 + Xß4)
= fa + Xp"ßl;
whence, remembering that
U = x2y2 + x2y2 + x3y3 + x4y4,
U = (oq -|- xßj (p atj + Xp"^) + (a2 + X/32) (p'a2 + Xp"ß2) + &c.,
and as a and ß are both on fl, we have,
?7= X (p + p") (a^ + a2ß2 + a3ß3 + a4ß4);
but since the line joining p' and p" is a generator of fl, the last factor must
vanish, and the line is therefore also a generator of U.
It is thus proved that U has four generators in common with fl
423. Verification of the Invariance of Intervene.
As an exercise in the use of the orthogonal system of co-ordinates, we
may note the following proposition :—
Let xlt x2, x3, xit and xlt x2, x3, x4, be two objects which are conveyed by
the transformation to ylt y2, y3, yit and y^ y^ yj, respectively, it is
desired to show that the intervene between the two original points is equal
to that between the transformed. The expressions for the cosine of the
intervene are
_________+ x.yx3 + x4x4
Oi2 + + x* + «/)! (x^ + + + >