of Edinhurgh, Session 1882 - 83 . 
157 
(p-p\==p{p-pdl>~\ 
( 17 ) < P'-P2=P(P-P2)1^~^’ 
^P - Pi- P{p-Pi)p~\ 
and by these relations the distances of p from the summits p\, p^, p\ 
of the displaced triangle are equal to the corresponding distances of 
p from the summits p^, p^, Pg before the displacement of the system. 
If we take the scalar of the product member to member, we get, 
owing to : 
(18) S(p - p\){p - p^{p - p'g) - + S(p - pi)(p - p^)(p - Pg). 
We shall refer again to this result when we have the correspond- 
ing result calculated by the second solution. 
By the second solution, we must satisfy the condition 
TV(cZp2 - dp^){dp -dp^) = 0, 
and as p represents any point of the system, it follows that all the 
relative displacements dp - dp^ must be parallel to one another. 
Let us designate by g the unit-vector in the direction of these 
displacements. We shall then have, when = o - p^, 
dP=-W 
= P + 
li'^ = /3^ + 2gS7j(3-g\ 
Applying (1), namely T/3' = T^, we get 
0 = ff{2Sr,IS-g}. 
Omitting g = 0, which corresponds to no displacement at all, we get 
dp==2g^gP, 
and 
(19) 13' = (3 + 2gSrjp ; 
or with the signification of f3 : 
P -Pi = p-Pi + ^^v(p - Pi) . 
From this, making successively p = P 2 , p = P 3 , and proceeding as in 
the case of the first solution, we establish, first the invariability of 
the triangle of which the summits are in p-^, p^, pg, before the displace- 
ment, and in p\, p^, p^ after the displacement. Secondly, we establish 
the three relations following: 
O 
