of Edinburgh, Session 1871-72. 
781 
cos ( m't + e 7 ); the period of any complex vibration is , and 
therefore at intervals of — ■ the configuration of the magnets 
will be the same to a spectator who changes the side from which 
he regards them in successive such intervals. Thus, if one magnet 
was originally at rest, the two will alternately be reduced to rest. 
When there are three equal magnets, it is easy to see that one 
fundamental mode is a swing of the whole as one piece, a second 
(if we suppose like or unlike poles adjacent to each other at each 
gap) is the middle magnet and the centre of inertia of the other 
two fixed, and the third has also the centre of inertia fixed, but the 
two extreme magnets are at each instant equally deflected in the 
same direction, while the middle one has a double deflection to the 
opposite side. It is troublesome, but not difficult, to think out the 
fundamental modes for four and even far five magnets ; hut it would 
be a waste of time to try it in that way for more. 
Generally if x r denote the displacement at time t of the rth 
magnet, and if we assume the masses, magnetisation, and gaps to be 
equal, we have 
x r + n 2 x r = fx + ^ “ (_ a + Xr+1 _ Xr y^ 
= ^ (xr - 1 + Xr+1 - 2av) , 
except for the ends of the series where r- 1 , and r = m, the number 
of magnets. 
Hence, multiplying by \ r and adding, we have 
where 
£ + = 0 , 
€ = % A r x r 
It will be sufficient to work this out for three magnets. Here, if 
we put -^-5 = e , we have 
war 
o L 
VOL. VII. 
