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On the Logarithmic Law of Atomic Weights. [Apr. 19, 



In the first section the reason is pointed out why the search for this 

 law has been fruitless, at least as hitherto pursued by the author. The 

 method he adopted was to plot down the atomic weights as ordinates 

 of a diagram of which the abscissas represented some simple numerical 

 series, and to endeavour to extract information from the resulting 

 curves. In this method atomic weights are represented by lines, the 

 ordinates of the figure. Now in the next section it will appear that, 

 in that case, the curve is represented by the equation — 



y = h . [log ( 2 x)P, 



and is further complicated by x not representing simple integer num- 

 bers, but a circular function of them. The search, therefore, by this 

 method was from the first hopeless, as the resulting curve is one which 

 has not been studied by geometers, and of which accordingly the 

 inquirer could not recognise the appearance when presented to him. 



In Section 2 another method is pursued. The successive atomic 

 weights, instead of being represented by lines, are represented by 

 volumes. A succession of spheres are taken whose volumes are pro- 

 portional to the atomic weights, and which may be called the atomic 

 spheres. When the radii of these spheres are plotted down on a 

 diagram as ordinates, and a series of integers as abscissas, the general 

 form of the logarithmic curve 



y = h log (qx) 



becomes apparent : and close scrutiny has shown that it expresses the 

 real law of nature. It is the central curve that threads its way 

 through the positions given by observation, and the deviations from 

 it -of the positions assigned by the actual atomic weights will be in- 

 cluded by making x a circular function of integer numbers, instead of 

 those numbers themselves. The first three terms of this function have 

 been determined. 



The issue of the investigation is to show that when such a diagram 

 is formed with ordinates which are the cube roots of the atomic 

 weights referred to hydrogen as unit, so that the ordinates may be 

 the radii of spheres whose volumes represent the atomic weights — 



1. The logarithmic curve — 



y m = h . log (ma), 

 (where log h = 0'785 



and log a = 1-986) 



threads its way through the positions plotted down from the observa- 

 tions . 



