Astronomy and High-speed Inertia. 83 



value ; whereas in the opposite half of the orbit, where it is 

 travelling against the sun's way, the inertia will be less than 

 the average. But in all cases the effective or apparent 

 inertia of the planet will be slightly larger than the mass on 

 which the force of gravity acts. Hence we may expect the 

 orbit to revolve in its own plane, or, in other words, the apses 

 must slowly progress ; for the effect will be much the same 

 as if gravity were proportionally diminished, the inertia 

 remaining constant. 



The theory of the apsidal angle in nearly circular orbits, 

 under a central force varying as any power of the distance, 

 is given by Newton with unexampled genius in Principia, 

 Book I, Section IX, and is thrown into orthodox analytic 

 form in Tait & Steele § 248. The result is that for a central 

 orbit subject to a law of force as the nth. power of distance, 

 the angle swept through by the radius vector between two 

 consecutive apses, i. e. between maximum and minimum 

 •radii, in orbits nearly circular, is 



7T 



Newton himself remarks that for a direct distance law this 

 angle will be a right angle; for a constant force, tt/\/3 ; 

 .and for the exact law of inverse square, precisely it ; thus 

 giving a perfectly repeated orbit without any apsidal 

 progression. But if the inverse square law were inexact, 

 the apsidal angle would differ from ir by a corresponding- 

 amount; and the direction of motion of the apses would be 

 progressive if gravity diminished faster than the inverse 

 square law, i. e. it* the index n exceeds — 2 numerically. 



To get the perturbed rate of progression for Mercuiy, 

 41 or 43 seconds of arc per century, — which value was 

 reckoned by Newcomb as the outstanding discordance from 

 theory, — we have only to remember that the planet makes 

 4 revolutions per annum, or 8 journeys from apse to apse, so 

 that in a century the discrepancy tt\ \/ (?> + n) —it has accu- 

 mulated 800 or more accurately 830 times ; so, to give the 

 observed value, 



1 43 



v/(3 + w; 830x180x3600 



Whence ?i= -2-00000016. 



This value was reckoned by Professor Asaph Hall in 

 Astr. J. vol. xiv. page 49, Boston 1894; but then there is no 

 other reason for supposing that the index is not exactly —2. 



G2 



