304 
MR. R. S. HEATH OH THE DYNAMICS OF 
Hence, the component angular velocity about a line (a l5 b x , c l5 f x , g x , h x ) of an angular 
velocity 6 about {a, b, c,f g, h) is 
+%+ cc x +ff x +gg x + hh x ) 
If the lines meet, this law is similar to the parallelogrammic law in ordinary space. 
To find the component angular velocity about any line, we multiply the given angular 
velocity by the cosine of the inclination of the line to the axis of rotation. 
Also the component translational velocity along the line (a x , b x , c x ,f v g x , h x ) due to 
the same angular velocity 6, is 
0(«/i +tyh + ch x +f<h+gb x + hc x ) 
Hence the expression af x -\-bg x -\-ch 1 -\-fa x -\-gb 1 -\-hc x denotes the velocity along one 
line due to a unit angular velocity about the other. It is sometimes called the moment 
of the two lines. 
33. We shall next find the rates of change of the coordinates of a line, due to the 
angular velocities o) x , a>. 2 , oj 3 , o> 4 , co 5 , co 6 . 
Take two points ( x, y. z, u), (x, y, z, u'), distant a right angle from each other. 
We already know expressions for x, y, z, u in terms of the oj’s. Now r 
{yz'-y'z) 
and therefore 
-yz—zy'+yz'—zy' 
— (xa) G —zco 4 —U(a. 2 )z —(—X(o 5 -fyoj 4 — ua s )y' 
+ (— ®'fi ) 6 + yo) 4 - u'co 3 )y — (x'co 0 —zw^—uo.^z. 
= o) G (xz'—x'z) — oj. 2 (uz '— u'z) + co d (xy'-\-x'y) — lo.^yu'—y'u) 
ct — oj 0 b -)- oj -C — oj 3 g-\- oj. 2 Ji 
Hence finally 
(.L — . ■——|— CO5C . — (t).yg oi.Jl 
b - OJgCi . —oj 4 c- fi ojg^* . -—c o-Ji 
c=—o) 5 a-\-(oJ) . — (oj+o) x g 
f— • — 0) 3 b+(0 2 c . —OJ G g-\-co-h 
g= (t) 3 a . — (ti x C-\- 0 ) G j , + 0 J 4 /i 
/i == — oj 2 a-{-oj x b ■ —ojg f 4$ 
Since —0 X , —0. 2 , &c., correspond to oj 1; co 2) &c., if we apply these formulae to the 
edges of the fixed tetrahedron we find 
