A Treatise On the Theory of Screws

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Hence we learn that 6 must intersect the axis of z at right angles. 6 is thus parallel to the plane of xy at a distance d ei = d ez = z, and accordingly we have the equations ( p g a) cos X z sin X = 0, (p 6 b) cos /j. Z sin /u, = 0, 215] FREEDOM OF THE FOURTH ORDER.
whence eliminating z and observing that X //, = 90 we obtain, p d = a cos 2 X + 6 sin 2 X, and eliminating p e, (b a) sin X cos X = z. If we desire the equation of the surface we have y = x tan X, and hence finally 225 Thus again w
...e 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 6 be any screw of the two-system, and accordingly the six co-ordinates of 9 must satisfy four linear equations which may be written If be a screw which intersects 6 at right angles, then we must also have inasmuch as these screws are reciprocal as well as rectangular.
From these six equations O l, ...


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