166 Mr J. Larmor, On Possible Systems of Jointed [May 26, 



which we started, which gives one condition, and the three other 

 independent circuits in the same layer give three more. Thus the 

 three layers give twelve conditions, which can easily be satisfied by 



the nine lines we started with, especially as three of them may be 

 removed by properly choosing the positions of the layers. 



Having thus proved the possibility of the arrangement, we 

 proceed as before to count all the independent relations of the 

 system, and find whether they are sufficient to fix it absolutely, — 

 or, if not, to find how many modes of deformation it possesses. 

 We project all the independent circuits on the axes, just as 

 before in the case of the binary system. There are 9 binary systems 

 contained in the ternary, 3 sets of 3 each ; but it will be clear on 

 consideration that the existence of 2 of these sets determines the 

 third set, which crosses them both. The independent circuits of 

 the ternary system are therefore those of these two sets of binaries, 

 and give equations 6 . 4 . 3 in number ; while the metrical relations 

 of the binaries give 6 . 4 conditions among the lengths of the seg- 

 ments, which are necessarily included in the former: so that there 

 are 6 . 4. 2 or 48 independent equations. There are also 27 rela- 

 tions between the direction cosines of the 27 lines, which are the 

 variables. Thus there are 75 equations in all between these 81 

 variable direction cosines. But the arbitrary axes introduce into 

 them 3 degrees of indeterminateness. There • are therefore still 

 3 degrees remaining : that is, the jointed system possesses three 

 degrees of internal freedom. 



And now the same considerations that we employed in the case 

 of a binary system show that we may introduce any additional 

 number of rods in each set, so that three rods shall meet at each 

 joint, when the system will still possess its three degrees of internal 

 freedom. 



This remarkable general result is in agreement with what we 

 can see to be true in particular cases. The simplest case of all 

 is that of a parallelepipedal system formed of three sets of parallel 

 jointed rods : here we can alter all the three angles between the 

 directions of the rods. Another simple case is that of a series of 



