508 MR, G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
When the obliquity is neglected, that equation may be written —= 1 + g jl — j, 
and it is proposed to find what values of n would make n equal to 12. 
In the course of the above investigation four different starting points were taken, 
viz.: those at the beginning of each period of integration. There are objections to 
taking any one of these, to give the numerical values required for the solution of the 
above equation ; for, on the one hand, the errors of each period accumulate on the 
next, and therefore it is advantageous to take one of the early periods; whilst, on the 
other hand, in the early periods the values of the quantities are affected by the sensi¬ 
bility of the solar terms, and by the obliquity of the ecliptic. The beginning of the 
fourth period was chosen, because by that time the solar terms had become insigni¬ 
ficant. At that epoch I found log n Q — 3 - 84753, when the present tropical year is the 
unit of time, and /x = '6659, g being the ratio of the orbital moment of momentum 
to the earth’s moment of momentum; also log s— 5 , 39378 —10, s being a constant. 
Now put x s =v = fl, and we have 
at — (l~b g)? + 7=0 
Then substituting the numerical values, 
ar*— 11727a + 40385 = 0 
This equation has two real roots, one of which is nearly equal to J/117 27, and the 
other to 40385-P 11727. By Horner’s method these roots are found to be 21’4320 
and 3’4559 respectively. These are the two values of the cube root of the earth’s 
rotation, for which the earth and moon move round as a rigid body. 
The first gives a day of 5 hours 36 minutes, and the second a day of about 
55-| m. s. days. 
The latter is the state to which the earth and moon tend, under the influence of 
tidal friction (whether of oceanic or bodily tides) in the far distant future. For this case 
Thomson and Tait give a day of 48 of our present days + the discrepancy between 
my value and theirs is explicable by the fact that they are considering a heterogeneous 
earth, whilst I treat a homogeneous one. Since on the hypothesis of heterogeneity the 
earth’s moment of inertia is about +\kr, whilst on that of homogeneity it is |-Mu 3 , and 
since the f- which occurs in the quantity s enters by means of the expression for the 
earth’s moment of inertia, it follows that in my solution g has been taken too small in 
the proportion 5 : 6. Hence if we wish to consider the case of heterogeneity, we must 
solve the equation x 4 —12664^+48462 = 0. The two roots of this equation are such 
that they give as the corresponding lengths of the day, 5 hours 16 minutes and 4O'4 days 
respectively. The remaining discrepancy (between 40 and 48) is doubtless due in part 
* ‘Nat. Phil.,’ § 276. They say:—“ It is probable that the moon, in ancient times liquid or viscous in 
its outer layer or throughout, was thus brought to turn always the same face to the earth,” In the new 
edition (1879) the ultimate effects of tidal friction are considered, 
