158 
Proceedings of the Boyal Society 
(p-p'i = P~Pi + ^V^v(p-Pi), 
(20) Ip'-P2 = P~P2 + ^V^v(p - P2) » 
* -P 3 = P- Ps + ^^v(p - P 3 ) ? 
by wbicli the distances of p from p\, p'g, p '3 are respectively the 
same as those of p from p^, pg, pg. 
We now take the scalar of the product member to member. This 
gives for the second member, 
^•{p-Pi){p-P2){p-P3) 
+ 2S.7]'^Y(p-p^){p-p^) Sy{p-p^), 
where represents the summing of the three terms obtained by the 
circular permutation of the indices. But this sum represents the 
vector 
7^S.(p-pi)(p-p2)(p-p3), 
and, treated by 2 S 77 , we get a term which reduces itself so as to 
give : 
(21) S. (p - p\){p - p'2)(p - p's) = - S. (p - pi)(p - p 2 )(p - P 3 ) . 
If Ave compare the sign in the second member of this result (21) 
with the sign in the corresponding result (18), we see at a glance 
that the displacements of the second solution produce a state of 
things which is incompatible with the conditions which constitute 
a physically rigid system, and that therefore the relation (18) alone 
characterises the systems which possess the complete rigidity from 
a geometrical point of view. 
As we have at our disposition two different expressions of the 
displacements according to the second solution, namely, the expres- 
sions (3) (with the values (11) of p and q) and the general expression 
(19), we will investigate into the properties of the second solution 
in using both expressions. But we will discuss first the properties 
of the first solution, and then, taking up the second, we will at the 
same time be able to show the correspondence (from a formal point 
of view) of the properties of both solutions. 
§3. 
By the first solution we have the expression (17), of AA^hich we 
Avuite again the first 
( 22 ) 
p' - p'l =P{P - Pi)P ■ \ 
