164 
Proceedings of the Royal Soeiety 
metrical in respect to the plane (38), and we vdll call therefore that 
plane i\\Q plane of symmetry. The component dr having displaced 
the point p into its position of symmetry p + dr, the component 
dp^ parallel to that plane (because perpendicular to rj), and being 
applied to every point, will shift the system of points p + dr out of 
their position of symmetry in causing them to move by a translation 
parallel to that plane, definite in direction and distance. We may 
designate the result of both the displacements dr and dp^ by the 
name of perversion. 
For the second solution we have also the expressions (3) which 
we may write now for^ and co-axial (by 11 ) and apply to 
a =Pi“Pc, ^ =p -Pc, 
a =Pi-p'c, = p -pe. 
We will now determine the angles u and v of respectively^ and g. 
Having 
]■ a'=^ap-‘, 
we get, by (5) and (20), 
(40) 
( c?a = 2 sin M . pY^a = 27jS7}a , 
t = 2 sin t? . qYl/3 = . 
Treating these relations by S . ^ we get, as the second members 
vanish, 
J S^tZa = 0 , 
t B^d/3=:0, 
(41) 
and consequently also 
Thus ^ must be in a plane perpendicular to We call ^ the 
unit-rector perpendicular to ^ and to 77 , so that 
and the system t], ^ will be trirectaiigular, ^ being also in the 
same plane with and to the left of it when seen from a point of 
the positive direction of rf. Let us take the origin of that system 
at the extremity of p^ in the plane of symmetry (38). 
In treating (40) by V.^, and remarking that Ip—p^, and that 
Y^pY^a—pYCY. we get (by a change of sign, in replacing 
V.Ca by V. aC) 
2 sin u . pY{CYal) = 2 ^S?^a , 
2Binv. qYaYpQ^2$Sy^p. 
