156 
Proceedings of the Boyal Society 
inent. For the sake of conciseness, we may designate tire points 
themselves by their vectors. We will also assume that the vectors 
Pj, p. 2 , &c., be given : the vectors p\, p'^, &c., p will be deduced from 
them according to the one or the other of the two solutions in 
question. 
Let us consider the groups of three points, p^, p^, p, where p is 
to represent (successively) any point of the system, save pj and pg. 
We put 
And we liken p-pi with /5, putting 
p-Pi = ^ . 
The origin of a and (3 will be at the extremity of p^, say in 0;^. 
Introducing, for any index, the notation 
p' = p + c7p, 
we shall have now 
da = dp 2 — dp-^ , 
d/3 = dp — dp ^ . 
We assume dp-^ to be given, and then the object of each of the 
two solutions will be the determination of dp2, dp^, &c. dp. 
The first solution, 
gives or g = But the double sign disappears in the 
operator q ( which becomes _p ( )2p\ We have therefore 
the same operator for all the displacements. Thus we get 
(U) p\-p\=P(P2-P^P'\ 
(1 5) p - p\ =p{p - 
Writing (15) for p = Ps, we have 
(16) P 3 ~ P i~ p(Pb~ Pi)P * 
The relations (16) and (14) establish the invariability of the 
triangle, which has its summity in p^, p 2 , pg, before the displacement, 
and, in p\, p 2 , p '3 after the displacement of the system. 
Looking on this triangle as a basis in reference to which the posi- 
tion of any other point of the system may be defined, we get by 
(15), and by subtracting successively (14) and (16) from (15), 
