of Edinhurgh, Session 1882 - 83 . 
159 
and which applies the same operator p ( ) ^ to any point of the 
system. Let us look upon ^9 as a datum. Moreover as 
Pi = Pi + ^Pi > 
p =p + dp , 
we assume also that the displacement dp-^ of the extremity of p-^ he 
given, so that the expression (22) will serve to the determination of 
the displacement dp of the extremity of p, which may be any point 
of the system, or also, in this case any point invariably connected 
with the points of the system of given points. 
Let us determine the point, or locus of points, of which the dis- 
placement is parallel to the axis ^ of p. If p^ designates such a 
point we define it by putting 
(23) = 
or 
(23 bis) dp^ = lt, 
where Hs a scalar to be determined. 
Applying the first of (17) we get for p = PqI 
(24) Po + &~ (pi + dp^) =p(pq - p^)p -^ . 
Transforming this by (5) and multiplying by p~^, we get 
By this equation S^(po - pf) remains indeterminate. We repre- 
sent this scalar by - s and add member to member 
-2=Sf(po-pi). 
Multiplying the sum by - ^ we get 
Taking first the scalar of both members gives 
(25) t + Btdp^ = 0, 
so that t is determined by this relation, and there remains 
( 26 ) p„ = f. + [p,+ 
By this expression we see that p^ represents the vector of any 
point of a straight parallel to I (z being an indeterminate scalar), 
and passing through a definite point, of which the vector depends 
on p and on p^ and dp-^, all the elements of which are given, or 
supposed to be given. 
VOL. XII. 
L 
