along connected Systems of Similar Bodies. 361 



The equations (20) are satisfied if 



2cos/3 + 2-n 2 = 0, 

 that is, if 



n = ZQO& — (30) 



In (29), (30) s may assume the m values 1 to m inclusive. 

 In the last case n = 0, and /3 = 7r ; and from (28), 



i/r r = — (— l)''cos nt. 



The equal amplitudes and opposite phases of consecutive 

 coordinates, i. e. angular displacements of the magnets, gives 

 rise to no potential energy, and therefore to a zero fre- 

 quency of vibration. In the first case (s=l) the angular 

 deflexions are all in the same direction, and the frequency is 

 the highest admissible. If at the same time m be very great, 

 n reaches its maximum value, corresponding to parallel 

 positions of all the magnets. If we call this value N, the 

 generalized form of (30), applicable to all masses and degrees 

 of magnetization, may be written 



n = N cos — (31) 



zm 



If m is great and s relatively small, (31) becomes approxi- 

 mately 



«= N 0-^> < 32 > 



so that as s diminishes we have a series of frequencies ap- 

 proaching N as an upper limit, and are reminded (as Fitz- 

 gerald remarks) of certain groups of spectrum lines. A nearer 

 approach to the remarkable laws of Balmer for hydrogen* 

 and of Kayser and Runge for the alkalies is arrived at by 

 supposing s constant while m varies. In this case, instead of 

 supposing that the whole series of lines correspond to various 

 modes of one highly compound system, we attribute each line 

 to a different system vibrating in a given special mode. 

 Apart from the better agreement of frequencies, this point of 

 view seems the more advantageous as we are spared the 

 necessity of selecting and justifying a special high value of m. 

 If we were to take s = 2 in (31) and attribute to m integral 

 values 3, 4, 5, ... , we should have a series of frequencies of 

 the same general character as the hydrogen series, but still 

 differing considerably in actual values. 



There is one circumstance which suggests doubts whether 



oev 



* Viz. ?i = N(l— 4m-2_), with m=3, 4, 5, &c. 



