Physical Significance of the Least Common Multiple. 711 



This represents, in my opinion, the only case in which the 

 L.C.M. of magnitudes affected by experimental error can 

 have an L.C.M. which is physically significant. Its charac- 

 teristic is that all the magnitudes are integral multiples o£ 

 some " natural unit " of which fractional values cannot 

 occur. If Prof. Baly's conclusions cannot be explained as 

 numerical coincidences, they must prove that there is an 

 indivisible unit of frequency in the absorption spectra which 

 he considers. But since there is, I believe, no general belief 

 at present in such a unit, it will be well to point out that the 

 measurements must fulfil certain conditions to be consistent 

 with that hypothesis. If they do not fulfil them, then, so far 

 as I can see, the agreements he finds must be numerical 

 coincidences, however small the apparent probability of such 

 a coincidence may be. 



If measured values a, b, c are to be integral multiples of a 

 natural unit and, at the same time, the statement is to be 

 significant that c is the L.C.M. of a and b, then it must be 

 possible to find an x such that, within experimental accuracy, 

 (a. b, c) = (p, q, r) . x, where p, q, r are integers and r is the 

 L.C.M. of p,q. 



This condition raises some difficulties when, as in Prof. 

 Baly's examples, the values obey his rule, being expressed in 

 units such that they are not integral. For consider a = 1'7, 

 /> = 5'6, c = 431'2. The only possible value of x is near to 

 0*1 ; but if #=0*1, the numbers of natnral units in each of 

 the measured magnitudes is 77, 56, 4312. And since the 

 only w T ay of proving the existence of a natural unit with any 

 satisfaction is by showing that measured values are in the 

 ratios of small integers, the mere fact that all Prof. Baly's 

 magnitudes must be represented, in terms of natural units, 

 by integers that are by no means small, makes it difficult for 

 him to establish from these experiments (I say nothing of 

 others) the only proposition that will make his results sig- 

 nificant. But once more, the only alternatives are (1) that 

 his agreements are mere coincidences, (2) that all experi- 

 mental physics is founded on a delusion and scales of notation 

 have physical importance. 



II. (By E. C. C. B.) 



Whilst minor objections might be raised to some of Dr. 

 Campbell's statements the main criticism he makes against 

 the least common multiple is obviously sound. I naturally 

 am in complete agreement with him in his conclusion that 

 the least common multiple of two different energy quanta 



