302 
MR. R. S. HEATH OH THE DYNAMICS OF 
From these it follows at once that 
— w i • Pi — w 6 ?n i “t" "h 0 • h = h 
In this way we arrive at the following equations :— 
4 — . — 0) 6 TOj + eOgWj 
m 1 = cogli . — ci>^n l 
Pi — 6Ji/i+co 3 mi + w 3 pi 
and similar equations hold for the other suffixes. 
29. Substitute these values in the equations for x, y, z, u, and make the variable 
tetrahedron instantaneously coincide with the fixed tetrahedron ; then 
x— . — co 6 y-\-a) 5 z —aqw 
y— co ti x . — co 5 z — o) 2 u 
z = — 0 ) 5 X-{- U)£) . — 0 ) 3 U 
'll — lOpC -j- OJ.^y -j- Ct)gZ . ^ 
To interpret the Hs suppose all except one (say co : ) to vanish. Then y, z vanish; 
therefore the distances of the moving point from the angular points 2 and 3 are con¬ 
stant. Hence the displacement is a rotation about the axis 23. If we put u= 1, we 
get£c=— &q. Hence io 1 is the angular velocity about the axis 23. This angular 
velocity is from the angular point 1, towards the angular point 4. 
Assuming the principle of superposition of small motions, we may say that the 
equations give us the rates of change of the variables x, y, z, u, due to a motion, which 
is the resultant of angular velocities aq, aq, aq, aq, co 5 , aq, about the six edges of the fixed 
tetrahedron, 23, 31, 12, 14, 24, 34 respectively. 
30. In exactly the same way we may express the variations of l, m, n, p, in terms 
of six other variables, 0 1} 0 2 , 0 3 , 0±, 0 5 , 0 e , defined as follows :— 
-"lPi"! 
i 
w aPi I 
0i = kk+ m i m * + n i>h+Pilh 
02=kk+ m -2 r> h+ n Ph+PPh 
03=hh+ m 3 m 4+ 
. 
°l= 44+ m 2 W 3+ '>h'>h+PlPA 
#5 = 44+ m ?Ph + Vi +P3 Vl 
6» 6 =44-f mpiii-P npi 2 +_pip 3 ^ 
