Chase.] "iO" [Oct. 6, 



24G. pendulum EsUni'ite of Moon's Mass. 



In Note 8, I anticipated slight modifications of my first estimate of 

 Moon's mass, as likely to be required by subsequent investigations. If 

 we apply the principles which are involved in the coefficient of solar tor- 

 sion. Note 162, to the determination of the length of Earth's theoretical 

 pendulum, we find 



=«(4) 



Z = ^ ^--^ = -^^ X (43082.05)2 -^ rr' = 1,142,882 miles. 



From this equation we deduce the relative value of Moon's mass, z^-, by 

 the proportion, 



p^ : I : : m^ : y, 



92,785,700 : 1,142,882 : : 81.1857 : 1 



This estimate differs from the one in Note 8 by less than -^\ of one percent. 



247. Rotation Estimate of Moon's Mass. 



The conviction, which I have often expressed (Note 220, etc.), that 

 rotation is only modified revolution, is further strengthened by the follow- 

 ing considerations. The orbital velocity {v^) which the combined energies 

 of Earth and Moon tend to give to an equatorial particle which is nearest 

 to the Moon, is about 2.18 times as great as the velocity (c^) which they 

 tend to give to the mean centre of gravity of Earth's oscillating particles. 

 The preponderating attraction of Earth prevents the action of these ten- 

 dencies, in any other way than as accelerating disturbances on the several 

 particles whose retarded and constrained revolution leads to axial rota- 

 tion. The greater acceleration, acting for a half-monthly oscillation {tj, 

 gives the mean orbital velocity of the system (^„), while the smaller accel- 

 eration, acting for a half-daily oscillation, gives Earth's equatorial velocity 

 of rotation («,), as is shown by the proportion 



18.4735 : .288188 : *: 14.7652942 v^-.^^^ 

 ?)^'= 2.1798 i'^. 

 If we designate the distances of the respective particles from the centre 

 of gravity of the system by d.^ and d^, we have d^ v^^ = d^ v^ ; d^ = 

 4.7514 d^. The theoretical mean intersections of d^ with Earth's sur- 

 face should be on the equator, and those of d^ should be on meridians, 

 but want of exact homogeneity, as well as orbital inclinations, maybe pre- 

 sumed slightly to modify their respective loci. The mean centre of gravity 

 of Earth's oscillatory particles is at the distance r from the surface, but 

 they are all also affected by wave-lengths equivalent to d^, so that we have 

 d^ =:d^-\-r = 4.7514 d^. Hence r = 3.7514 d^ ; d^ =.26657 r = 1056.35 

 miles ; dp = 5019.15 m.; r — d^ = 2906.45 m.; m^ + ^ = (238,869 -r- 

 2906.45) fj. = 82.1858 fx; mj = 81.1858 a, a vafue which corresponds ex- 

 actly with the one in the foregoing note. 



