Triple Pendulums iciih Mutual Interaction. 31 



write, as sufficiently expressing this initial state, 

 For£ = 0, a!=JC Q =$(y Q ), !/=y , 2=*o = ^(j/o), j 



^ =0 <k ah >m 



a ' dt u ' dt~ J - j 



Then, by substituting (39) in (30), we obtain as the solution 

 for this initial state in terms of a? , y , and z the following : 



6x = 2(2ic Q — y — z ) cos m 2 t + 2{x + y + z ) cos m d t, 



fy = Kyv — ~o) cos mtf — (Zxo—yo—Zo) cos m 2 £ 



+ 2(«b + y + '«'o)cosm 8 «, )>(40) 

 6#= — 3(j/ — ^0) cos m^ — (2.v — y Q —z ) cos ??? 2 £ 



+ 2(# + */o + z )cos>m 3 £. 



IV. Mechanical and Electkical Systems. 



In the case o£ the interconnexion of three dissimilar 

 vibrating systems nine constants may be seen to be mathe- 

 matically possible. (See equations (l)-(o), p. 612, Phil. 

 Mag. Nov. 1920, and (11) of the present paper.) But in 

 the electrical case it is at once apparent that the mutual 

 inductances reduce the number of their numerical values 

 from six to three, each occurring twice. And in the appro- 

 priate form of the equations for the mechanical case the 

 same reduction must hold. Hence only six constants are 

 possible as a maximum in either case. 



If now the three vibrating systems, when isolated, are 

 made precisely alike, two more constants are lost, the total 

 number possible under these circumstances being reduced 

 to four. Thus, in equations (l)-(3) (p. 612, Phil. Mag, 

 Nov. 1920) on omitting subscripts from the L's and S's 

 the four constants would become 1/LS, Mj/L, M 2 /L, and 

 M 3 /L, the first of these four occurring on the left side of 

 th^ equations and the other three on the right. 



In the mechanical case the terms due to the coupling are 

 not confined to the right side but affect also the variable 

 •occurring on the left. Hence in the mechanical case now 

 under consideration, with masses of bobs and total lengths 

 of pendulums all equal, it is not surprising to find that two 

 constants, / and A, appear on the left side, leaving only 

 two, k and /, on the right. Thus we have four constants in 

 all, as in the somewhat analogous electrical case. But the 

 latter shows a superiority in point of symmetry. Thus 

 the mechanical case in question attains the same generality 

 as the electrical case it imitates, but displays this generality 

 in a less simple fashion. 



