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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410] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 449
(VII) The departure between two ranges cannot be zero unless the ranges
are coincident, or unless the departure between every pair of ranges in that
star is also zero.
(VIII) Of the ranges in a star, two (distinct or coincident) are at infinity,
i.e. have each an infinite departure from all the remainder.
(IX) An infinite range has an infinite departure, not only with every
range in its star, but with every range in the extent.
(X) If the several ranges in one star correspond one to one with the
several ranges in another, and if the two infinite ranges in one star have as
their correspondents the infinite ranges in the other; then the departure
between any two ranges in one star is equal to that between the two corre-
sponding ranges in the other.
409. The Form of the Departure Function.
The analogy of these several axioms to those which have guided us to
the discovery of the intervene, shows that the investigation for the function
of departure will be conducted precisely as that of the intervene has been;
accordingly, we need not repeat the several steps of the investigation, but
enunciate the general result, as follows :—
Let xlt x2, and ylt y2 be the co-ordinates of any two ranges in a star, and
let Xi, Xs, and g2 be iAe co-ordinates of the two infinite ranges in that star.
Then the departure between (xlt x2) and (ylt y2) is
(yig2 — y-ipi)
(.yih^ — y2h-i) — x2ffi)
410. On the Arrangement of the Infinite Ranges.
Every star in the extent will have two infinite ranges, and we have now
to see how these several infinite ranges in the extent can be compendiously
organized into a whole.
To aid in this we have assumed Axiom IX., the effect of which is to
render the following statement true. Let several objects on a range, 0, be
the vertices of a corresponding number of stars. If 0 be an infinite range
in any one of the stars, then it is so in every one.
Let alt a2, as be any three ranges in an extent. Then every range in
the same extent can be expressed by
ÆqCl'l ^2^2 ‘^•3^3 j
where xlt x2, xs are the three co-ordinates of the range. It is required to
determine the relation between xlt x2, x3 if this range be infinite.
B.
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