176 



Mr. G. H. Darwin. 



[Jnne 19, 



if the moon-eartli system were started with less than of its actual 

 moment of momentum, it would not be possible for the two bodies to 

 move so that the earth should always show the same face to the moon. 

 Again if we travel along the line of momentum there must be some 



point for which yx^ is a maximum, and since yx z — ^- there must be 



some point for which the number of planetary rotations is greatest 

 during one revolution of the satellite, or shortly there must be some 

 configuration for which there is a maximum number of days in the 

 month. 



Now y:& is equal to x^Qi—x), and this is a maximum when x — ^i 

 and the maximum number of days in the month is (f/0 3 (Ji — f A) or 

 3 3 -, 



— /V ; if h is equal to 4, as is nearly the case for the homogeneous earth 



and moon, this becomes 27. 



Hence it follows that we now have very nearly the maximum 

 number of days in the month. A more accurate investigation in my 

 paper on the " Precession of a Viscous Spheroid," showed that takiug 

 account of solar tidal friction and of the obliquity to the ecliptic the 

 maximum number of days is about 29, and that we have already 

 passed through the phase of maximum. 



We will now consider the physical meaning of the several parts of 

 the figures. 



It will be supposed that the resultant moment of momentum of the 

 whole system corresponds to a clockwise rotation. 



Now imagine two points with the same abscissa, one on the 

 momentum line and the other on the energy curve, and suppose the 

 one on the energy curve to guide that on the momentum line. 



Then since we are supposing frictional tides to be raised on the 

 planet, therefore the energy must degrade, and however the two 

 points are set initially, the point on the energy curve must always 

 slide down a slope carrying with it the other point. 



Now looking at fig. 1 or 2, we see that there are four slopes in the 

 energy curve, two running down to the planet, and two others which 

 run down to the minimum. In fig-. 3 on the other hand there are 



o 



only two slopes, both of which run down to the planet. 



In the first case there are four ways in which the system may 

 degrade, according to the way it was started ; in the second only two 

 ways. 



i. Then in fig. 1, for all points of the line of momentum from C 

 through E to infinity, x is negative and y is positive ; therefore this 

 indicates an an ti- clockwise revolution of the satellite, and a clockwise 

 rotation of the planet, but the m. of m. planetary rotation is greater 

 than that of the orbital motion. The corresponding part of the curve 

 of energy slopes uniformly down, hence however the system be started, 



