152 
Proceedings of the Royal Society 
enabled to deduce the supplementary condition by which the rigidity 
of a geometrical system shall be established in all its completeness. 
Both these expressions are reducible to the same definite form, each 
expression yielding properties which correspond each to the other 
in the two kinds of displacement. To the line representing the 
axis of rotation in the screw-rotation corresponds the plane repre- 
senting the plane of symmetry in the perversion ; to the translation 
in the first, parallel to the definite axis of rotation, corresponds a 
translation in the second, parallel to the definite plane of symmetry. 
With the expressions combined of the two kinds of displacement 
we shall be able to establish a general demonstration of the pro- 
position according to which two given successive perversions are 
equivalent to a determinate screw-rotation. 
§ 1 - 
Let ns consider first the displacements of two points A and B, 
supposing that they displace themselves to A' and B' respectively. 
Assuming an arbitrary origin for the vectors of these points, 
we put 
OiA=a, OiB-^, 
C\A' = a', OiB' = /?'. 
By onr hypothesis we must have 
I Ta' = Ta , 
(1) ]t/3'=T/3, 
(T(^'-a') = T(/J-a). 
We transform the third of these equations, by the help of the two 
first, into 
(2) Sa'jS' - Sa^ - 0 . 
The most general solution of the two first equations may receive 
the form 
where and ^ may be any versors whatever. 
By these expressions the first member of (2) becomes 
- Sa/3. 
Let ns call Z this expression, so that 
Z = 0 
will represent the equation (2). 
