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LXIIL The Physical Significance of the Least Common Mul- 

 tiple. By Norman Campbell, Sc.I)., and E. C. C. Baly, 

 C.B.K, M.Sc, F.R.S. * 



Note. — This paper is the result of a long private discussion 

 between its authors. They have resolved some of their 

 differences and misunderstandings, but since a residuum of 

 disagreement remains they think it well to express their 

 views in the form of a criticism and a reply. 



I. (By N. R. C.) 



1. IN interpreting his work on absorption spectra Prof. 

 X Baly {e.g. Phil. Mag. xl. p. 1, 1920) has based 



some of his conclusions on the observation that one of the 

 frequencies (or wave-numbers) characteristic of the spectrum 

 is the least common multiple of other characteristic fre- 

 quencies. So far as I can discover, this is the first time 

 that physical significance has been attributed to the L.C.M. 

 of measured magnitudes ; the occasion seems appropriate to 

 examine what that significance may be. 



2. The conception of an L.C.M. is usually applied only 

 to integers ; Prof. Baly applies it to fractions, still meaning, 

 of course, by the L.C.M. the least common integral multiple. 

 No objection can be taken to this extension in general ; any 

 fractions, so long as they are commensurable, have an L.C.M. 

 which is given by 



T n A/T - L.C.M. of numerators of the fractions ,.,. 



H.C.F. of denominators of the fractions ' 



where H.C.F. means highest common integral factor. But 

 it may be observed at once that the conception is applicable 

 only if the fractions are commensurable ; there cannot be 

 any L.C.M. of the length of the side of a square and the 

 length of its diagonal. In speaking of an L.C.M., Prof. 

 Baly is assuming that the real frequencies characteristic of 

 an absorption spectrum are commensurable. Though no 

 arguments can be raised against this assumption, I believe 

 that none can be raised for it ; there can be no arguments 

 until relations are found for absorption spectra such as arc 

 known for the " hydrogen-like " emission spectra. The fre- 

 quencies of the Balmer series are doubtless commensurable ; 

 but it is still doubtful whether the frequencies of the similar 

 spectra in other elements are so. However, Prof. Baly 

 might well claim that the success of his interpretation 



* Communicated by the Authors. 



