306 
MR. R. S. HEATH ON THE DYNAMICS OF 
called a twist about a certain screw whose axis is the given line. The same motion 
may be effected by a twist on a screw whose axis is the polar line. Let the angular 
velocities about the edges of the fixed quadrantal tetrahedron due to any such twist 
be Oa, Ob, He, Of, fig, Oh, respectively, where 
a 2 +b 2 +c 3 +f 3 +g 2 +h 3 = 1. 
Then a, b, c, f, g, h, may be defined to be the coordinates of the screw, and O the 
magnitude of the twist about it. The quantity 2{af-)-bg+ch} may be called the 
parameter of the screw, and will be denoted by k. If k= 0, the coordinates are the 
coordinates of a line, and the motion degenerates into a rotation about the line 
represented by the coordinates. 
The locus of lines which have no lengthwdse velocity due to a twist about the 
screw, is the linear complex 
sf-\-hg-\-oh-\-fa-j-gb + he = 0. 
This we shall call (after Lindemann) the Rotation Complex. If Oa, Ob . . . had 
been translations instead of rotations, the same property would have been enjoyed by 
the polar complex 
an■+b^ fi- cc +f f-\- gg+hh = 0. 
This is called the Translation Complex. Any property of one complex has a 
corresponding property of the other. 
The trajectory of every point on a line of the rotation complex is perpendicular to 
the line. Hence the trajectory of any point whatever is normal to the plane which 
corresponds to the point in the complex. 
If we refer back to the expressions for a, b, c,f, g, h, we see that 
a/’+ bg+c/i+fet+g6 + he = 0 
and also 
a« + hb + cc+ ff-\- gg+hh = 0 
Hence, if any line belongs to either complex, it will belong to it after receiving a 
small twist about the corresponding screw. Hence, by superimposing such small 
twists, it follows that both complexes are transformed into themselves by a twist about 
the corresponding screw. 
36. The screw may be replaced by two rotations, or two translations, in an infinite 
number of ways; any line may be taken as one axis of rotation or translation, the 
other axis being the conjugate polar of the first with respect to the Rotation Complex. 
For let the twist about the given screw be equivalent to rotations X, g, about axes 
a x , b x , c x , f x , g x , h v and a, 2 , b. 2 , c 2 , f 2 , g%, h 2 , respectively. 
