Cause of the Structure of Spectra, 249 



The values of m just given were calculated out with the 

 ^expectation that they would give evidence of nodal sub- 

 division connected with the vibrations. If we consider a 

 vibrating circle, we may take a vibration whose wave-length 

 is twice the circumference of the circle as the most natural 

 fundamental mode, and then its associated harmonics will 

 have their nodes at angular intervals ^ir/s round the cir- 

 cumference, where s is an integer. But the circle has still 

 another type of harmonic, which we may call undertones, if 

 those just given are overtones, because it is possible for 

 q times the circumference to be half the wave-length of a 

 permanent vibration of the circle, which may have harmonics 

 such that tbe angular interval between successive nodes is 

 2irq\s, and the expression 'lirqjs^ where g and s have any 

 integral values, may be taken as representing all the possible 

 modes of vibration of the circle. Now many of these different 

 nodes will coincide with one another if the starting-point of 

 all the main vibrations in the circle is the same. With this 

 point as origin, the nodes whose angular intervals are 1/s, 

 2tt+1/s, Aw + l/s .... will coincide, and there will be a 

 .tendency for the corresponding modes of vibration to increase 

 one another's stability in comparison with that of modes 

 having isolated nodes. On this account the modes of vibra- 

 tion which we should expect to find most pronounced in 

 a circle under circumstances of perfect freedom would be 

 those whose wave-lengths are represented by such an ex- 

 pression as r±l/s, where r and s are both integers. These 

 considerations explain the principle on which the above table 

 was originally drawn up, namely, that the fractional parts of 

 m in the table should be mostly of the form ±l/s, repre- 

 senting the simplest nodal subdivision, and that the value 

 s = '2 should be of most importance, then * = 3 and s = 4, and 

 so on. Ultimately we shall see how the vibrating circle is a 

 helpful model. Let us then select from the table those 

 values of in which contain fractions near *5, namely, 



4-518 5-525 7'510 8-518. 



The fractions in their departure from the exact value *5 

 show a certain periodicity s which can be more clearly seen 

 if we write the numbers thus :— 4 '5 + '018. 5*5 + '018 + '007, 

 (6-5 + -018) 7-5 + -018--007, and 8'5 + '018. 



At first I was fairly satisfied with the tabulated numbers 

 as evidence of nodal subdivision, but in view of the sharpness 

 with which the integral values oOOO, 4'000, and so on, occur, 

 I had begun to think them less convincing, till on reading 

 the recent happy discoveries of E. C. Pickering and of 



