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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349]
THE THEORY OF SCREW-CHÄINS.
383
obtain the following equations by simply adding the equations just written
for each separate screw
2^ = (11) 20,+(12) +
= (nl) 2Ö, + (w.2) S02... + (nn) 20}l.
If the Bynames in the first system neutralize then their components on
the screws of reference must vanish or
20,= 0, ... 20» = 0.
But it is obvious from the equations just written that in this case
S^ = 0,...2</>„ = 0,
and therefore the corresponding Dynames will also neutralize.
Given n pairs of corresponding Dynames in the two systems, we obtain
n? linear equations which will be adequate to determine uniquely the n2
constants of the type (11), (12), &c. It is thus manifest that n given pairs
of Dynames suffice to determine the Dyname in either system, corresponding
to a given Dyname in the other. It is of course assumed that in this case
the intensities of the two corresponding Dynames in each of the «-pairs are
given as well as the screws on which they lie.
349. Properties of this correspondence.
To illustrate the distinction between this Dyname correspondence and
the screw correspondence previously discussed, let us take the case of two
cylindroids. We have already seen that, given any three pairs of corre-
sponding screws, the correspondence is then completely defined (§ 343).
Any fourth screw on one of the cylindroids will have its correspondent on
the other immediately pointed out by the equality of two anharmomc ratios.
The case of the Dyname correspondence is, however, different inasmuch as
we require more than two pairs of corresponding Dynames on the two
cylindroids, in order to completely define the correspondence. For any
third Dyname 0 on one of the cylindroids can be resolved into two Dynames
and 0.2 on the two screws containing the given Dynames. These com-
ponents will determine the components on the corresponding cylindroid,
which being compounded, will give </> the Dyname corresponding to d.
It is remarkable that two pairs of Dynames should establish the corre-
spondence as completely as three pairs of screws. But it will be observed
that to be given a pair of corresponding screws on the two cylindroids is
in reality only to be given one datum. For one of the screws may be
chosen arbitrarily; and as the other only requires one parameter to fix it