160 
Proceedings of the Poyal Society 
Subtracting (24) member to member from (22) in which we re- 
place p p + dp, and p\ by p-^ dp-^, we get 
(27) dp = lt + dcr, 
where we put 
(28) «?o- = b(p-Po)y'‘-(p-po)]- 
The displacement dp is therefore represented by two components. 
The first, which is parallel to ^ is the same for all the points, 
because (27) gives, by S^c7o- = 0, 
- t, 
and, by (25), the value of t depends on the data alone, and is there- 
fore a constant. In fact, by this last relation we have the series of 
equations 
(29) -t = BCdp^ = ^^dp.^ = = &c., 
which show that the displacements of all the points of the system, 
when projected on the direction ^ give one and the same projection, 
so that we may continue to designate by dp^ as by (23 bis). 
The component da- is perpendicular to because, owing to ^ 
we have 
SCp(p - Po)p-' = ^p~n(p - Po) = SC(p - Po)- 
Hence, 
S.^f?o- = 0. 
This relation assigns to do- a plane perpendicular to As to the 
direction of da- within that plane, we must deduce it from (28). 
By its definition, da- represents the displacement of p — pQ after a 
conical rotation round the axis parallel to I, which passes through 
the point p^, and which therefore is the straight line (26) repre- 
senting the locus of pQ ; the angle of rotation being 2u, namely, twice 
the angle of p, when 
p = cos u i- 1 sin u . 
We may transform da- in many ways. As, for example, putting 
P-Po^^y 
we have 
(30) do- = pa-p~'^ - a- . 
Applying (5), we get 
(31) do ^2 sin u . pY^o . 
