Fuly 29, 1886} 
But before entering on the new ground I think a few words 
of explanation will be necessary. Mr, Darwin takes exception 
to a proposition of mine in that it holds his theory to be depen- 
dent upon the genesis of the moon at the present surface of the 
earth. I was led to this conclusion chiefly from the apparent 
stress laid on the condition by the writer on that subject in 
Thomson and Tait’s ‘‘ Natural Philosophy,” but as the author 
now states that his theory is not so founded, I think it ought to 
be allowed that it is not. But I think the argument can be 
put in another way ; for if the moon beallowed to have separated 
at a period over four hours, the theory would be at variance 
with the calculations of its author (for he fixes the period at 
between two and four hours). A flock of meteorites is proposed 
as a form in which the moon might have receded from the earth. 
Nothing can be gained by this, for the flock of meteorites can- 
not come nearly so close to the earth as the moon ina single 
mass, without the constituent members being separated and each 
compelled to describe an independent orbit with its own period. 
In other words, the tidal force would separate the flock of 
meteorites at a greater distance than it would the single body. 
And at a greater distance it is not necessary, as the moon in her 
conglomerate form could not be objected to, 
Also, he quotes and questions a passage, which is to the effect 
that two heavenly bodies cannot revolve about their centre of 
inertia with their surfaces nearly in contact, unless one is smaller 
than, and denser than, the other by a certain amount. The 
case was intended for where the two bodies move as parts of a 
rigid body, 7.e. each keeps the same face towards the other ; 
but I omitted to insert this condition in giving the rule. 
Coming now to the results of my second investigation. In 
his last reply, as well as in several other places, Mr. Darwin 
advances the Martian system as affording a striking confirmation 
of the influence of tidal friction. The view is that the extreme 
minuteness of the inner satellite has preserved it as a standing 
memorial of the primitive time of rotation of Mars round his 
axis (see the Observatory, July, 1879). 
Now I think it must have been taken for granted that the 
smallness of the satellite would allow the above state of things 
to come about, for an estimate of the comparative effects pro- 
duced on Mars by solar and by satellite tidal friction, and the 
reaction on the satellite shows that, according to the estimated 
dimensions of the latter body, its period must be considerably 
more disturbed than that of Mars. Prof. Newcomb estimates 
the diameter of the inner satellite at from 10 to 40 miles. If we 
take the lower estimate and suppose the body to be only as 
_ dense as the sun, then its mass will be 86000" times smaller than 
the sun’s. But the distance of the satellite—6000 miles—is 
23333 times less than the sun’s distance, and this number must 
be cubed to get the effect on the tidal force through greater 
proximity. After making the proper allowances, as above, it 
will be found that the satellite has a tidal force fifty times less 
than that of the sun. Ifthe tidal retardation or acceleration, 
as the case may be, varies as the square of the tidal force, as 
Mr. Darwin allows, then 1/2500 of the planet’s retardation must 
be counteracted by the satellite tides, which go round in the 
_ reverse direction to the solar tide. Here the reaction on the 
satellite must be considerable, for an approximate calculation 
will show that the orbital momentum of that body is only about 
1/2,800,000 of the planet’s rotational momentum. It will not 
be necessary to work out the calculation. Suffice it to say that 
the density and dimensions of the planet are taken from New- 
-comb’s tables, and that the distribution of density is supposed to 
be like that of the earth, giving a rotational momentum equal to 
that of homogeneous density multiplied by 0°83. 
If solar tides lengthen the Martian day by one minute, then 
the rotational momentum will have been reduced by about 
1/1400 of the whole, and the satellite must have produced an 
effect in the opposite direction 2500 times as small, so that the 
actual effect of the satellite is to increase the rotational momen- 
tum by 1/3,500,000. And since the reaction on the satellite 
will be equal and in the opposite direction, more than half its 
momentum (which is 1/2,800,000 of the planet’s) will be lost, 
which will reduce its distance to the surface of the planet. 
Hence we are led to the startling conclusion that before solar 
if tidal friction can alter the rotation-period of Mars by one 
minute, the inner satellite must fall into the planet. I have not 
_ taken into the calculation the circumstance that the satellite tide 
_ goes round quicker than the solar one, nor that, as the satellite 
approached the planet, its tides would increase, the purpose here 
__ being only to give an account of the relative changes that should 
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NATURE 
287 
take place at about the present configuration. Further, it 
would seem that solar tides could not have reduced the period 
of Mars much, even if it be supposed that the inner satellite was 
first at any greater distance, for then it must either have gone 
out and attained a longer period than Mars, or it must have 
fallen into the central body long ere this. There seems no 
escape from these conclusions, unless the little body gives out an 
extraordinary amount of light for its size (for its probable size is 
judged by its brightness), but this seems so improbable, that it 
would be unreasonable to suppose so. As for the density, the 
inferior limit must be not less than half that allowed, otherwise 
the tidal force of the planet would break the body up. 
Now a few words may be said on the future history of the 
moon. Before, I have said that the tracing back of the moon 
has apparently been carried too far in one direction ; and now I 
think that the tracing forward, supposing tidal friction to have 
free play in the future, has been carried too far in the other 
direction. According to Thomson and Tait (‘‘ Elements of 
Natural Philosophy”), the moon’s distance should increase to 
347,100 miles by the time the earth’s rotation relatively to the 
moon is stopped, when the two bodies should revolve in about 
48°3 days. Now it would take all the rotative momentum that 
the earth would lose during the change to send the moon to the 
above distance ; or, in other words, if there were no solar tides 
or other causes to prevent all the rotational momentum lost 
by the earth going to increase the moon’s orbital momentum, 
the moon’s distance would not be increased to beyond the above. 
But it is clear that a considerable portion of the rotative energy 
would be lost in solar tidal friction, which would have no part 
in increasing the moon’s distance from the earth. For the moon 
to recede to the distance named, the earth must not only have 
its present moment of momentum, but also as much as solar 
tides would extract during the interval. At the present time 
the retardation through solar tides is not a small fraction of the 
whole, and it should increase till, at the other end of the jour- 
ney, it will be more than half the whole retardation, for then 
the solar will be greater than the lunar tide. 
I believe that Messrs. Darwin and Ball, who wrote a year or 
two before the date of Thomson and Tait’s work, give the dis- 
tance to which the moon will recede a; even greater than the 
above, and say her period of revolution will be about fifty days. 
Certain remarks made by Mr. Darwin iu the larger work of 
Thomson and Tait leave some doubt as to whether a correction 
has recenily been made in the moment of momentum of the 
earth’s rotation, but even if the earth be supposed homogeneous, 
the rotational momentum would not be sufficient to send the 
moon to the above distance, when allowance is made for solar 
tidal retardation. Hence it seems that no allowance has been 
made for the effects of this agency, and that when such allow- 
ance is made, the moon’s destination must fall far short of the 
estimates given. If the distance (347,109 miles) be reduced to 
about 320,000 miles, I think it would be nearer the mark. 
Mr. Darwin says that the eccentricity of the lunar orbit, the 
obliquity of the ecliptic, and other elements would be co- 
ordinated together by supposing that the moon first had a 
separate existence at no great distance from the present surface 
of the earth, and with small differential motion with respect 
thereto. I will only say that the case is so complicated, and 
the data so unreliable, that the results of the calculations in- 
volved seem to be little better than guess-work. 
As for the distribution of satellites in the solar system, I think 
the majority of diverse theories would hold that they should be 
more numerous far from the sun, for the simple reason that solar 
disturbance would be less there. James NOLAN 
Dergholm, Victoria, May 25 
Mr. NoLan is correct in supposing that I made no nume- 
rical calculation with regard to the inner satellites of Mars. I 
accept the calculation which he gives, and admit that the 
present period of revolution of the satellite cannot be regarded, 
as I supposed, as a memorial of the primitive period of rotation 
of the planet. 
I see, however, no reason, as yet, to recede from the follow- 
ing statement (Pil. Trans., Part I. 1880, p. 883), which em- 
bodies the essence of the argument, without the erroneous 
phrase :-— 
‘Tt is here (in the case of Mars) alone in the whole system that 
we find a satellite moving orbitally faster than the planet 
rotates. This will also be the ultimate fate of our moon, be- 
cause, after the moon’s orbital motion has been reduced to 
