708 Dr. N. Campbell and Prof. E. C. C. Baly on the 



justified his assumption, and therefore no further objection 

 on this score can be taken. 



3. The point to which I wish to draw attention is that the 

 L.C.M. given by (1) is determinate only if the fractions are 

 known with perfect exactitude ; it is quite indeterminate 

 if there is any experimental error whatever. In this the 

 L.C.M. differs from the functions which are most frequently 

 used in physics. Thus, if we are concerned with the product 

 of two magnitudes A and B, an uncertainty of (say) 1 per 

 cent, in their values produces an uncertainty of only about 

 1 per cent, in their product. But the same uncertainty, or 

 any uncertainty whatsoever, produces an infinite uncertainty 

 in their L.C.M. If we do not know the values exactly, 

 then there is an infinite number of values covering an 

 infinite range, any of which may be the L.C.M. 



For there is an infinite number of ratios p/q (where p and 

 q are prime to each other) which differ by less than any 

 assigned amount from each other ; whatever our experi- 

 mental accuracy, there is an infinite number of pairs p, q 

 the ratio of which can be used with equal right to represent 

 the measured value. Let these ratios for one of the measured 



values be ^, ^ 2 . . • for the other —,—.... Then, by 

 (1), the L.C.M. of — and - will not be equal, or even 



qi s'i 



approximately equal in general, to the L.C.M. of — and — .. 



92 S 2 



If (p l5 q u r 1? s-j) are all prime to each other then the L.C.M. 

 is piVx, and will differ enormously according to the approxi- 

 mation adopted. Thus, if we cannot measure a magnitude 

 to 1 in 1000, we cannot have any reason to adopt the 

 approximation 22/7 rather than the approximation 355/113 ; 

 but if we take the L.C.M. of this magnitude with (say) 

 31/10, the two results we shall get will be 682 and 11005, 

 which are in the ratio of 22 to 355. Their ratio is not that 

 of any small integers ; they are magnitudes as completely 

 different as any magnitudes can be. 



4. This is so obvious that, if we had been in the habit of 

 expressing fractions in the vulgar form, the indeterminate- 

 ness of the L.C.M. of any fractions of which the value is 

 not known with complete mathematical accuracy would have 

 been immediately apparent. It is only if we use, as we 

 always do, decimal notation (a term which will be employed 

 to denote also similar notation based on a radix other than 

 10) that there is any appearance of determinateness. For 

 when we adopt such notation, we fix the denominator of all 



