318
THE THEORY OF SCREWS.
[298,
Let
Po'/’l — a@l = -3^1;
Pe'/’s ~ b03 = Äs,
Pe(t>s~ cds = Xs>
Pecl)2 4" ^^2 — X3f
Pe^i + ^4 = Xit
pe<ps + C0S = Xs,
and the equation becomes
0 = («, + as) (X, + X,) + (a, + a4) (X3 + X4) + (a6 + a6) (X, + X6);
and the two other L equations give
° = (Ä + Ä) (X3 + Z2) + (ß3 + ft) (X3 + Z4) + (ft + Ä) (X5 + X8),
0 = (yj + y2) (X-L + Za) + (y3 + y4) (X3 + Z4) + (y6 + 76) (X5 + Xs).
It we eliminate X3+X3, X3 + X4, Xs + Xe from these equations, we
should have
0 =
oti + a2
Ä+/3,
71 +
a3+ a4
Ä + Ä
Va+ 7t
a5+ a6
ßs + ße
7i + Vs
But this would only be the case if a, ß, 7 were parallel to a plane, which is
not generally true. Therefore, we can only satisfy these equations, under
ordinary circumstances, by the assumption
JTj 4- X3 = 0, X3 + X4 = 0, Xs + Xs = 0.
In like manner, the equations of the M type give
Pø^a<f> PoL^Ør] 0>
Pe^ßt — Pß^el — 0>
— Py^ei — 0.
Substituting, in the first of these, we get
+ Pe (««101 — aa2</>2 + 6a3</>3 — 6a4^>4 + ca5</>5 — ca6</>6),
— pa (ayidi - at)302 + bT]303 — byjiOt + - erlös') = 0;
which reduces to
aajJTi — a«sJT2 4- ba3X3 — &a4X4 + ca3X3 — ca3Xe = 0;
but we have already seen that Xj + 2l2 = 0, &c., whence we obtain
XT (aa. + aa2) + X3 (ba3 + 6a4) + X3 (ca5 + ca6) = 0;
with the similar equations
Xt (aß, + aß2) + X3 (bß3 + bß4) + X5 (eßs + eße) = 0,
((171 + ay3) + X3 (by3 + by4) + X6 (cy5 + cy6) — 0.
These prove that, unless a, ß, y be parallel to a plane, we must have