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of rotations in S, about planes all passing through one point are 
equivalent, then these systems themselves are equivalent. 
Corollary. The elementary rotation in S, can be reduced in an 
infinite number of ways to a pair of conjugate rotations correspon- 
ding to the conjugate rotations of its section with an arbitrary Ss. 
The reduction of a pair of (perfectly) normal planes comes to the 
same thing as the 
ProBLEM: To indicate among the pairs of conjugate axes of the 
elementary motion in S3 the pair which determines with a given 
point in S, a pair of normal planes. 
It is a matter of course that we have but to search among those 
conjugate axes that cross each other normally. 
Lemma. The locus of the point of intersection of the normal planes 
through a pair of lines crossing each other normally is the circle 
having the common normal as a diameter and the plane of which 
is normal to the S3 through those lines. 
If namely in fig. 1 
a and a' are the given 
rectangular lines, 2 
their common normal, 
O the point of inter- 
section of a pair of 
normal planes through 
a and a’, the plane 
(On), must be normal 
to the two lines a and 
Fig. 1. a’ (to a because it 
contains zn and QA’, to a because it contains x and OA). The 
plane (On) is therefore normal to the S3 («a’). 
Hence for the proposed reduction, only those conjugate normal 
axes of the section are to be taken into account whose common 
normal passes through the projection O' of the centre of motion 0; 
this is according to a wellknown property the normal from O' on 
the central axis ¢ of the sectional motion. 
When in the above figure the lines a and a’ represent a pair of 
such axes, their common normal cuts the central axis in the point 
C in such a way that AC X CA’ =p’, p being the speed of the 
screw motion about e. The locus of the point of intersection of the 
normal planes through «a and a’ consequently passes through a 
fixed point P, in the normal ot S; through C and at a distance 
