along connected Systems of Similar Bodies. 359 



if n represent the value of n appropriate to k = 0, i.e. to 

 infinitely long waves. Here n = 0, when \=2a. In this 

 case yfr r+ i = — \jr r . 



Fitzgerald considers, further, a more general linear system 

 constructed by connecting a series of equidistant wheels by 

 means of indiarubber bands. " By connecting the wheels 

 each with its next neighbour we get the simplest system. If 

 to this be superposed a system of connexion of each with 

 its next neighbour but two, and so on, complex systems with 

 very various relations between wave-length and velocity can 

 be constructed depending on the relative strengths of the 

 bands employed." If the bands may be crossed, the potential 

 energy takes the form 



K = iy 1 (^±^_ 1 ) 2 +i7i(^±^+i) 2 



+ .-, (17) 



which is only less general than (5) by the limitation 



iO o ±C 1 ±O 2 ±...=0 (18) 



Prof. Fitzgerald appears to limit himself to the lower sign in 

 the alternatives, so that C in (10) vanishes, This leads to 

 (12), from which his result differs, but probably only by a 

 slip of the pen. 



If we take the upper sign throughout, (8) becomes 



-f» 2 =C 1 cos 2 ^+C 2 cos 2 ^ + C 3 cos 2 ^ + ... . (19) 



It may be observed that Prof. Fitzgerald's system will 

 have the most general potential energy possible (5), if in 

 addition to the elastic connexions between the wheels there 

 be introduced a force of restitution acting upon each wheel 

 independently. 



As an example in which C 2 is finite as well as C 1? let us 

 imagine a system of masses of which each is connected to its 

 immediate neighbours on the two sides by an elastic rod 

 capable of bending but without inertia. Here 



P = ... + lc(2^ r _ 1 -A/r r _ 2 -f r ) 2 +l C (2^ r ->/r r _ 1 -f r+1 )2 



A comparison with (5) gives 



C =Qc, C l = 4c, C 2 = — c, 

 so that 



C = C -2C 1 -2C 2 = 0. 



