166 
Proceedings of the Royal Society 
where a and c constitute two more arbitrary elements : in all, six 
arbitraries, all of them entering in both the expressions (43) and 
(44). 
It is true that in the case of one perversion one might define ^ 
so as to render dp^ parallel to it, and so we would have a = 0, and 
there would be only five arbitraries ; hut if we want to combine two 
given perversions, we are not free to dispose of the direction 
of I arbitrarily. 
§ 5 . 
The results of the two preceding paragraphs will now he applied 
to the demonstration of the proposition according to which two per- 
versions applied in succession to a given system of points produce 
a displacement represented by a screw-rotation. 
This proposition may he looked upon as known already, hut its 
demonstration in its greatest generality by the quaternion method 
has perhaps not yet been given. 
W e assume that the planes of symmetry corresponding to the two 
perversions have been determined and brought under the form 
S^T + C = 0, SVr +C' = 0. 
We assume also that pc? p ’c taken so as to assign to p\ a 
position on the line of intersection of the two planes. Let more- 
over pi, p, represent the vectors of two points given arbitrarily 
before the displacement, and let p\ p^ be the vectors of these points 
after the first perversion and p\, p" the vectors of the same points 
after the second perversion. Then putting 
a = Pi - Pc , a = p'l - p'o, a' = p'\ - p\ , 
/^ = P-Pc5 /5'=P -p'e, -Pfy 
we may represent the perversions by 
a =pap~^ , a ' ^pdp "^ , 
P' = ql3q~\ = • 
Hence if we put 
P'P = Piy = 
a" =l)ia^r' , 
r=si/3?r’- 
we have 
