1884.] Wickerwork, and their Degrees of Internal Freedom. 163 



by three such sets of six placed one above the other, and tied 

 together by nine rods passing through them, making twenty-seven 

 rods in all. 



Consider then a system of six, represented on the flat by the 

 annexed scheme, in which their direction cosines to fixed axes 

 l l m 1 n 1 ,..., Xj/^i/,,... are indicated, and also the lengths of the three 

 segments of each a 1 , b v c x (=a 1 + 6J, ... a,, @ 1} 7, (= a t + /3J,... By 

 expressing that the projections of each independent circuit on the 

 axes of co-ordinates are zero, we exhaust all the independent 

 relations of the system. We thus obtain four sets of three equa- 

 tions each, of which the following is one : 



a x m x - a 2 m 2 = aji t - a 2 fj,A , 



«A - <V* 9 = a i»i - Va J 

 and we have in addition six equations of the type 



P + *»■+'«* = 1. 



Thus we have 18 equations between 18 variables. But these 



variables involve 3 indeterminates, depending on the directions of 



the axes : and as we know that the system is not rigid, there is a 



fourth indeterminate. Therefore the 18 equations are equivalent 



to only 14 independent equations, and that can only be by reason 



of the existence of 4 relations between the coefficients, i.e. between 



the lengths of the segments of the rods. And, conversely, if we 



obtain these four relations independently, we can infer that the 



jointed system is not rigid. 



A 

 We can readily obtain them as follows: — Let a 1 ? 1 denote the 



angle between the lines a r , a l : then by equating two expressions 



for the square of the diagonal of the reticulation ap.fi^\ we obtain 



A A 



a x 2 + a/ — 2a 1 a 1 cos a^ = a* -f a 2 2 — 2a 2 ot 2 cos a 2 % 2 . 



A A 



or a 1 a 1 cos a 1 i 1 — a 2 a 2 cos a 2 <z 2 = 1 (a x 2 + a x 2 — a 2 2 — a 2 2 ), 



and similarly 



b 3 B 3 cos &> 3 - b& cos 6 2 /3 2 = 1 {of + /3 3 2 - J 2 2 - #/), 



A A 



c l7l cos c l7l - c 3 7 3 cos c 3 7 3 = \ (c x 2 + 7l 2 - c 3 2 - 7s 8 ), 



three equations between the cosines of the angles a x i x , b 2 fi 2 , c 3 y 3 . 



But we know that these angles are not determinate, therefore the 



• A 



result of eliminating cos 6 2 /3 2 between the two first equations must 



be equivalent to the third. That result is 



