312 Hon. R. J. Strutt on the Tendency of the 



There are various ways in which the problem may be 

 formulated, and which of these should be adopted is to a 

 great extent a matter of judgment. There is no unique 

 measure of the tendency to approximate to whole numbers, 

 although the phrase is a convenient one for general purposes. 

 The exact measure depends on what kind of approximation 

 we regard as most striking : whether, for instance, it would 

 be thought more remarkable that a few atomic weights should 

 be almost exactly whole numbers, while others departed largely 

 from the nearest whole number, or that all should be fairly 

 close, though none may be extremely so. 



In Prof. Mallet's paper, referred to above, the proportion 

 of the well-ascertained atomic weights, eighteen in number, 

 which deviate from the nearest whole number by less than *1 

 is taken as the fundamental datum. This method of treatment 

 has advantages, but is in some respects very arbitrary. There 

 is nothing to fix the particular limit of *1, except a con- 

 sideration of the numbers themselves. Another way of 

 regarding the matter is to take the sum of the deviations 

 (without regard to sign) of the atomic weights from the 

 nearest whole numbers, and to calculate the probability that 

 this sum should be as low as it is observed to be, supposing 

 that the atomic weights had been fixed by chance alone. 

 There seems to be less that is arbitrary about this method of 

 formulating the problem than about any other. 



The individual atomic weights cannot deviate by more than 

 a fixed amount, *5, from the nearest whole number. What we 

 require is the probability that after a given number (i) of 

 " trials " the sum of the results should not exceed a certain 

 given amount, as ; the result of each trial lying between 

 and *5, and any value between these limits being equally 

 likely. 



This problem is solved by Laplace (Theorie Analytique 

 des Probabilites, p. 259). It follows from the result there 

 given that the probability required is 



^{(iHl-'J+mi-*)'-...}. 



The series is to be continued only so long as the quantities 

 raised to the power i are positive. 



Laplace applied the solution to calculate the probability 

 that the sum of the inclinations of the planes of the planetary 

 orbits to that of the sun's equator should be, by chance only, 

 so small as it is observed to be — a question closely analogous 

 to that under discussion, since the inclination cannot in any 



