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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38 THE THEORY OK SCREWS. [41-
41. The Canonical Co-Reciprocals.
If all the six screws of a co-reciprocal system are to pass through the same
point, they must in general constitute a pair of screws of pitches 4- a and
— a on an axis OX, a pair of screws of pitches 4- b and — b on an axis 0 Y
which intersects OX at right angles, and a pair of screws of pitches + c and
— c on an axis OZ perpendicular to both 0X and 0 Y.
It is convenient to speak of a co-reciprocal system thus arranged as a set
of canonical co-reciprocals. The three rectangular axes OX, OY, OZ we may
refer to as the associated Cartesian axes.
It a,, a2)... a6 be tho six co-ordinates of a screw referred to the canonical
co-reciprocals, then the pitch is given in general by the equation
pa = a (a/ - a22) + b (a32 - a4-') + c (a52 - a62).
It must be remembered that in this formula we assume that tlie co-ordi-
nates satisfy the condition § 35
1 = («! + a.,)2 + (as + a,)2 + (a5 + a6)2.
Of course this condition is not necessarily complied with when at, a2, ... or
some of them are infinite, as they are in the case of a screw of infinite
pitch § 44.
In general the direction cosines of the screw a are
ai + «2> «a + a4> a5 + a6.
42. An Expression for the Virtual Coefficient.
Let X', p, V be the direction cosines of the screw 0 (of pitch pY) which
passes through the point x, y, z. Let X'', p”, v" be the direction cosines of
the screw a (of pitch pa) which passes through the point x", y", z". Then it
can easily be shown that the virtual coefficient of 0 and a is half the
expression
I x' — x", y’ — y", z' — z"
(p» +P4>)(X’X''+ p'p" + p'v")- X' p! t v .
X" , p" , v"
43. Equations of a Screw,
Given the six co-ordinates ... a6 of a screw, with reference to a set
of six canonical co-reciprocals, it is required to find the equations of that screw
with reference to the associated Cartesian axes.
If we take for 0 in the expression just written the screw of pitch a in the
canonical system, thus making
= 1; p = 0; v = 0; x = 0; y = 0; z’ = 0,