A Treatise on the Theory of Screws
Forfatter: Sir Robert Stawell Ball
År: 1900
Forlag: The University Press
Sted: Cambride
Sider: 544
UDK: 531.1
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295] DEVELOPMENTS OF THE DYNAMICA.L THEORY. 313
and of X'
aj cos O' + /3j sin O', ... a6cos O' + ß6sin Ö'.
In like manner, let p and p' be two screws on a cylindroid, of which the
two principal screws are p and
Let , ' be the angles which p and p' make respectively with £.
Then the co-ordinates of p are
cos + sin . r/a cos + & sin (f>,
and of p
cos ' + £ sin ye cos ' + £6 sin ', &c.
We shall now suppose that the two cylindroids a, ß and p, % are so
circumstanced that the latter is the locus of the impulsive wrenches cor-
responding to the several instantaneous screws on the former with respect
to the rigid body which is to be regarded as absolutely free. We shall
further assume that p is the impulsive screw which has X as its instantaneous
screw, and that the relation of p' to X' is of the same nature.
If, however, the four screws X, X.', p, p' possess the relations thus indi-
cated, it is necessary that they satisfy the conditions already proved (§ 281).
These are two-fold, and they are expressed by the following equations, as
already shown: —
—cos (Xp) + —7*', , cos (Xp') = 2-ctååS
cos (Xp) cos (X p )
Pk _ Pk
cos (Xp) p cos (X'p) ’
We shall arbitrarily choose X' and p, so as to satisfy the conditions
CT A'p = 0, W xp' = 0,
and thus the second of the two equations is satisfied. These two equations
will give O' as a function of , and ' as a function of 0. We can thus
eliminate 0' and ' from the first of the two equations, and the result will
be a relation connecting 0 and . This equation will exhibit the relation
between any instantaneous screw 0 on one cylindroid, and the corresponding
impulsive screw on the other.
It will be observed that when the two cylindroids are given, the required
equation is completely defined. The homographic relations of p and X is
thus completely determined by the geometrical relations of the two cylin-
droids.