Gravitational Attraction on Chemical Composition. 253 



the Newtonian coefficient. But it is in the earth-moon 

 system that a departure from strict uniformity of coefficient 

 would be most readily detected ; it is shown below that, if 

 the gravitational indices (§ 4) for earth and moon differed 

 by one part in 20,000,000,, the fluctuations of the moon's 

 longitude would comprise a term of period one lunar month 

 and amplitude one second of arc. Hence it follows that the 

 earth and her satellite, widely as they differ in mean density, 

 must have, within one part in many millions, the same 

 gravitational index. It should be possible to speak more 

 definitely when the short- period terms in the moon's lon- 

 gitude have been adequately discussed ; and this question, 

 I gather, is engaging the attention of Prof. E. W. Brown, 

 who has lately * expressed his belief " that sensible fluc- 

 tuations with periods comparable with a month also exist/' 



I. When it is not assumed that the Newtonian coefficient 

 of gravitation is universally constant, the simplest form which 

 the law of attraction between homogeneous masses m, m' can 

 take is given by the expression 



yy'mm'r~ 2 ... ... (1) 



for the attracting force, where 7, y' depend on the nature 

 •of the bodies m, m' respectively, and ?*is the distance between 

 those bodies. (It is understood that the term mass as here 

 used is equivalent to inertia.) It will be convenient to call 

 -7, 7' the gravitational indices of m, m', and 77' the Newtonian 

 ■coefficient for that pair of substances. To revert to the strictly 

 Newtonian law of gravitation, we have only to suppose that 

 y 2 = y' 2 =. . ., each of these quantities being then identical 

 with the Bewtonian constant, which has now become the 

 Newtonian coefficient for all pairs of substances. 



0. Let the masses of" sun, earth, and moon be M, m lz m 2 

 respectively, and their gravitational indices G, 7^ 70. Then 

 the mean index of the earth-moon system is 



{m l y l + m 2 y 2 )/(m l + m 2 ), 



and the excesses of 7^ y 2 respectively above this mean are 



( + m 2 , — m l ){y i — y. 2 )/(m 1 -\-m 2 ). ... (2) 



In lunar theory attention is ordinarily restricted to the 

 mean index, which is taken to be the same for earth and 

 moon ; that is to say, any possible distinction between y x and 

 -72 is disregarded. Our special problem is to find how the 

 motion is modified when 71 — 72 does not vanish, so that the 



* Month. Not. R. A. S. Ixxiii. 9 (Suppl. 1913) p. 694. 



