362
THE THEORY OF SCREWS.
[333,
will have
(§ 326)
its corresponding impulsive screw
defined by the equations
+ eary = p2 + a2 ; — eay2 =
+ ebys = azn - bz3 - (2; - ebyt =
+ ec% = cy0 - ay0 - l22; - ecye = - ay0- cy3 -122.
By substituting these in the equations just given, we obtain
pi ~ a2;
az3 + bz3 -l32;
az0 - bza -I»
a2 - pi2
a
b
a
2
— az0 - bz0 + l32
al±Pl+B’
a
a2 - Pi3
a
cy0 - ay0 - l-2
a1 ~ Pi
a
aP + pi2 , B„
a
c
ay^ + cy„ -4-I?
aa - Pi
a
a
c
In like manner from the screw on U
0, 0, 1,
0, 0, 0
we obtain
+ a-j/j = az0 - bz0 - l32;
+ bys = pi + ;
- cry = bx„ - cx0 — I2;
- ay2 = — az3 — bz0 - l3;
- byt = p2 - b2;
- c-rf = bx0 + C«o — I2.
for ->), we
Introducing these into the equations
azy-bzy-ls2 B
have
&+ps_A
b ~A
az0 + bz3 4-12
a
b‘‘~P2--A'
b ~A
(izq • bzü l32
a
„ az0 + bz0 + li2
a
— cx0
c
aza — bz0 —ls2
+ B‘
a
az,, + bz0 + ls2
a
a
— bx„ — cXf, + (2
c
(iZq bz0 l32 (izq 4- bZfi -f~ l3
a a
Thus we have eight equations while there are nine co-ordinates of the
rigid body. This ambiguity was, however, to be expected because, as proved
in § 306, there is a singly infinite number of rigid bodies which stand to the
two cylindroids in the desired relation.
The equations, however, contain one short of the total number of co-
ordinates ; æ0, y0, zQ, (2, I.2, l32, p2, p2 are all present but p2 is absent.