300 
MR. R. S. HEATH OH THE DYNAMICS OF 
Again, since a 4 =/j, b$=g Y , c 4 =/q, &c., the last three of the original equations are 
the same as the first three, with f, g, h written respectively for a, b, c. Hence these 
three give rise to the group of equations 
+9(^i + k)~Hh+ m i)—' a {h +Pi)=0 
—/( w s+%) • +h(l. 2 -\-m l )—b (m 4 +y> 2 )=0 
/(m 3 +w 2 )-^(w 1 + l 3 ) . -c(n 4 +_p 3 ) = 0 
a (h ~hPi) + ^ { m i +P 2 )+ c { n 4 +P 3 ) • =0 
It is not difficult to show that the first three equations of the first group, with the 
last of this group, contain all the rest. 
Besides these we have the equation 
af+bg+ch=0. 
If we substitute for f, g, h, their values in terms of a, b, c, it becomes 
ik+Pi)^ ,c {( n 4 ! +Ps){k+' in i) — ( m 4 +P 2 )( w i+y}+ two similar terms=0. 
This with the equation 
u{h+Pi) + &(™ 4 +#,) + c(n 4 +p 3) = 0 
will give us a quadratic in the ratios of a : b. 
Hence there are two lines which remain fixed during any displacement. Since the 
equations to find f g, h are exactly the same as those to find a, b, c it follows that one 
of the lines will be the polar of the other. 
The above work implies that the determinant 
h 3 
c i 3 
(( 2 5 ^2 1 3 
C 2 3 
a S 3 
h 3 
c 3 I3 
vanishes. I have verified by actual calculation and reduction in terms of (l, m, w,p), L2i3i4 
that the determinant and all its first minors vanish; but the work is too long to be 
reproduced here. 
Expressing the fact that two lines remain fixed during any displacement in kine- 
matical language we learn that any displacement whatever of a rigid body may be 
produced by two rotations, about two lines which are conjugate to each other with 
reference to the absolute. Instead of rotations we might have said translations; or, 
expressing one of the rotations as a translation along the polar line of its axis, we 
learn that any displacement of a rigid body may be effected by a rotation about a line 
