162 Proceedings of the Poyal Society 
round its axis by an angle 2u. The pitch h of the screw being 
The displacement, according to the first solution, is therefore 
appropriately called a screiu-rotation. 
When the angle of rotation is infinitely small, then the versor p 
in the expression (31) reduces itself to unity, and if we designate 
by e the instantaneous axis of rotation ^ x we get by (27) and (31) 
(33) dp = €li + Vc(p - pq) . 
In the expression (27) of the finite displacement dp, we may con- 
sider now the data to be the six following scalar elements ; — 
The two which determine ^ ; the two which determine the position 
of the axis of the screw by the point where it intersects a plane 
perpendicular to ^ passing through the origin 0 of the vectors p, 
this point being 
V(VCPo-OorV.(CV.poO, 
(the part - ^S^Pq disappearing in the expression of dp) ; and, finally, 
the angle u and the value of t, forming the last two elements. 
Likewise the expression (33) depends on the three scalar elements 
contained in e, on the two which determine V(V^Pq. Q, and finally 
on h, six elements in all. 
§ 4 . 
For the discussion of the second solution we will write the first 
of the expressions (20) under the form 
(34) dp - dp^ = 2t^Si^(p - Pi) . 
We will look upon dp^ and as the principal data by which to 
determine the displacement dp of any point p, when of course p^ 
and p are given. 
Let us determine the point p^, or locus of points, for which the 
displacement is perpendicular to t]. The condition will be (analo- 
gous to (23)) for p = p,, 
(35) Sr]dp, = 0. 
Writing (34) for p = Pe, we get 
(36) dp,- dp-^^ = 27]Sr){p,- p^). 
Treating this by S . we get, by (35), 
(37) -Srjdp^^ -2Sr][p,-p^); 
