January 17, 19 18] 



NATURE 



jin ratio to the velocity of light c, each of these terms 



■••cuines 



;« = ;«o ( I - — j = w„ sec 0, 



\Mi' 11 >iii/j represents* the ratio r/c-. 



Astronomical Applications. 



Since inertia is a function of speed, it becunus a 



question whether some astronomical perturbations may 



not thus be produced and accounted lor. This problem 



I attacked in the Philosophical Magazine for August, 



191 7. It is true that the motion of planets is slow 



compared with the speed of light, but it is immensely 



quicker than that of cannon-balls or of any artificial 



movement that we can cause on earth. Moreover, the 



fleets, if any, may perhaps turn out to be cumulative, 



li it is well known that the position of planets has 



i\v been observed for some centuries with prodigious 



■curacy. 



The quickest moving planet is Mercury, and since it 

 tkes four journeys round the sun every year, there 

 some reasonable chance of perceptible accumulation 

 '■ small effects in a moderate time. Now there is 

 known to be an interesting historical outstanding dis- 

 crepancy in the motion of Mercury which the theory of 

 gravitation fails to explain. 



The orbit of any planet or satellite subject to a per- 

 turbing cause, such as the attraction of a third body, 

 was shown by Newton to rotate in its own plane, the 

 position of its perihelion changing slightlv at each revo- 

 lution. In most cases gravitation can account for the 

 hole of this progress of perihelion ; but the orbit of Mer- 

 ry had been by careful measurement proved to revolve 

 ine forty or, more carefully estimated, forty-three 

 onds of arc per century more than could be accounted 

 1 by any known gravitative perturbation. It is not 

 much, but it is reckoned unmistakable — no one ques- 

 tions the fact — and many attempts have been made to 

 explain it. 



Leverrier invented an intra-mercurial planet, Vulcan, 

 account for this progress of the perihelion of Mer- 

 cury's orbit; but no such imaginary planet has ever 

 been seen. Other astronomers have surmised that the 

 law. of gravitation might be slightly inaccurate; or, 

 again, that the force of gravity travelled at a finite 

 speed. Recently Einstein has' applied the theory of 

 relativity to the problem, and by extremely complex 

 reasoning has arrived at the required result. 



It remains to see whether without any of those 

 efforts the straightforward and simple electrical theory 

 of matter cannot account for the observed progression. 

 Hitherto the attemot has been made to tamper with 

 the force acting on the planet ; we now leave the force 

 alone and tamper with the planet's inertia, as in- 

 creased by its motion through the aether, and varied 

 by any variations in that motion. 



The whole solar system is known to be travelling [ 

 among the stars; and sometimes the motion of a planet : 

 as it revolves round the sun will agree in direction 



s Whfn velocitv is constant, as it is during purely transvers- or rentrip-fal ac- 

 celeration, the effective or transverse inertia is simply wo«ec 0, being greater j 

 than the slow speed or rest inertia in the inverse ratio /^' li -^ ), as stated 



above ; but when velocity is increasins or decreasing by reaion of a longi- 

 tudinal force, we can write the conditions thus : 



/H = 7//o>:ecp 

 urn ~ til ^ tan /S 



-and F = ''^^>=,»,o^sec2^.g 

 = wo sec* /sj' 



so that hi^h-speed longitudinal inertia i« «/o fee' ^, ard is greater than the 

 slow speed or rest inertia in the ratio sec-' p. or, what is the same thing. 



ban the transr 



NO. 2516, VOL. 100] 



and is also, curiously, greater than the transverse inertia at the same speed, 

 in the ratio sec- iS. 



with a component of the solar drift, while at other 

 times — i.e. in the other half of its orbit — the planet's 

 orbital motion and a component of the solar drift will 

 be in opposite directions. Thus the absolute or re- 

 sultant speed of the planet througli th(> aether will 

 var>-, and hence, on the electrical tlnnx of matter, 

 its effective inertia will vary too. 



It remains only to calculate what the effect of this 

 varying inertia will be, given any reasonable value for 

 the sun's true motion through the aether of space. 



The resultant speed of the planet is to be reckoned as. 

 v'(i'=' + V'-f2T;Vcos0), 

 where ^ is the angle made by its motion v, at any 

 instant, with V, the solar drift. This last has a com- 

 p>onent ^ in the plane of the orbit, such that cos^ = 

 tx)s A cos h, 6 being the longitude and A the latitude 

 of the sun's true way referred to the direction of the 

 orbital motion v. So, expressing mass as a function 

 of velocity in Ihe ordinary equation of particle dy- 

 namics for any central force, 



where u is written for i/r in ordinary polar co-ordin- 

 ates, the mass will depend on phase, and will be found 

 to contain a factor i-fcos6. 



Introducing this factor due to varying inertia into 

 the above differential equation, I found it to take a 

 form familiar to electricians, viz. : 



.r -}- (cr + n\v = E cos //, 

 or, rather, a special case of this, with k = o and n = p. 

 In other words, it represents the case when free and 

 forced vibrations are of exactly the same period, and 

 undamped; it is the equation of perfect resonance. 

 The solution accordingly shows a steadily increasing- 

 amplitude, without limit, as time goes on, 



x== — t.smnt. 



In the same way the astronomical problem exhibits 

 accumulation or resonance as regards progress of peri- 

 helion, the perturbation being essentially synchronous 

 with the phases of orbital revolution ; and accordingly 

 after the lapse of, say, a century, the minute perturba- 

 tion due to fluctuating inertia, even though so small 

 as one-tenth of a second per revolution, may have 

 accumulated in the course of a century to the still small, 

 but very perceptible, value of forty-three seconds of arc. 

 Moreover, the kind of perturbation caused by fluctuat- 

 ing inertia, as expressed by the equation worjced out in 

 the August, igiy, Phil. Mag., turns out to be exactly the 

 kind of perturbation required, viz. a revolution of the 

 orbit in its own plane ; and it will be of the right value 

 provided the true or real solar drift ha« a component 

 equal to twice the earth's orbital velocity in a direction 

 parallel to the minor axis of the planet's orbit. 



The progress of f>eriheHon of a planet's orbit, after 

 n revolutions, comes out, according to this simple 

 theory, 



, TT/n'V cos * 



where v is the average speed of the planet, and e 

 the small eccentricity of its orbit; the unknown solar 

 drift is V, in a direction making an angle * with the 

 minor axis of the orbit ; and c is the velocity of light. 

 Assuming a drift of the above value, such as is re- 

 quired for Mercurv, I proceeded to try its effect on 

 Mars, and, as is shown in the August Phil. Mag., found 

 that it caused Mars's perihelion to revolve seven 

 seconds of arc per centurv ; which, I learn, is con- 

 sidered by astronomers to be the outstanding dis- 

 crepancy for Mars. 



