215] FREEDOM OF THE FOURTH ORDER. 225
whence eliminating z and observing that X — /z = 90° we obtain,
pä = a cos2 X+b sin2X,
and eliminating pe,
(b — a) sin X cos X= z.
If we desire the equation of the surface we have
y = æ tan X,
and hence finally
(b — a) xy = z (a? + y1').
Thus again we arrive at the well-known equation of the cylindroid.
We can also prove in the following manner the fundamental theorem
that among the screws belonging to any two-system there are two which
intersect at right angles (§ 13).
Let 0 be any screw of the two-system, and accordingly the six co-ordinates
of 6 must satisfy four linear equations which may be written
-41^1 + • • • + A60e — 0,
B& + ...+ Ba06 = O,
Ci^i + ... + Cfßs = 0,
+ ... + D606 = 0.
If <£ be a screw which intersects 0 at right angles, then we must
also have
^S + -- + ^S=°>
p101^i + --- +2>606</>8 = O,
inasmuch as these screws are reciprocal as well as rectangular.
From these six equations 0lt...0e can be eliminated, and wé-have the
resulting equation in the co-ordinates of
dÆ dR dR dR dR dR = 0.
dfa’ dcf>2’ dcf>3 d<t>t ’ dcf>5 d<l>6
Pi<l>i, P-&, Ps>^3, p^t, P^> P^c,
Al, -åø. A3, -4.4, a6, Ae>
Bi, Bi, B3, Bi, b6, Ba
c» c2, ot, Ci, C5, Gti
A, ds, Ds, Di, d5, De
B.
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