170 Frequency Ranges of JSf on-generating Force. 



If Z n denotes the determinant obtained by taking the terms 

 common to the first n rows and columns of either (4) or (5), 

 when all the as except cl x are zero, 



Z n = {l-(2^1)\}Z n _ l -a 2 Z B _^ 



/ n \ 2 

 where z = f — J . Hence i£ z is a root of Z n _! = 0, and if the 



corresponding root of Z„ = differs from z by a small 

 quantity s, then approximately 



a 2 Z„ _9 i 



, wnen z = z . 



This affords a ready method of testing the approximation 

 given by z . 



To the degree of accuracy shown in the table the last three 

 lower limits follow from Z 3 = 0; the rest of the roots exhibited 

 are given by Z 2 = 0, with the small correction s if necessary. 



6. The cumulative effect of nongenerating force of long 

 period, being comparatively feeble, is likely to escape notice, 

 except in the case of a system in which the motional re- 

 sistance is inappreciable and the time of observation extended. 

 As both of these conditions hold for the solar system, it is 

 natural to inquire whether any outstanding discrepancies 

 between astronomical calculations and observations could be 

 traced to the neglect of such slow cumulative action. Records 

 of solar eclipses are available over a long period, and it is 

 found that the calculated paths exhibit a regularly varying 

 error which becomes more marked with the remoteness of 

 the eclipse *. The regularity of variation is held to confirm 

 the accuracy of the record; and the question remains as to 

 how the discrepancy arises. 



I would suggest that it may be due to the cumulative 

 action of some lunar disturbance of relatively long period — 

 an action requiring for its detection a higher approximation, 

 perhaps, than has been attained in existing calculations. 



* See, for example, " The Moon's Motion," Nature, Oct. 1908, p. 599. 



