472 
MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
The numbers are such that — is expressed in degrees per million years. 
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The various values which — is capable of assuming as the viscosity and obliquity 
vary is best shown graphically. In Plate 36, figs. 2 and 3, each curve corresponds 
to a given degree of viscosity, that is to say to a given value of e, and the ordinates 
• dx 
give the values of — as the obliquity increases from 0° to 90°. The scale at the side 
of each figure is a scale of degrees per hundred million years— e.g., if we had e=30° 
and i about 57°, the obliquity would be increasing at the rate of about 3° 45' per 
hundred million years. 
The behaviour of this family of curves is so very peculiar for high degrees of 
viscosity, that I have given a special figure (viz.: Plate 36, fig. 3) for the viscosities 
for which e=40°, 41°, 42°, 43°, 44°. 
The peculiarly rapid variation of the forms of the curves for these values of e is due 
to the rising of the fortnightly tide into prominence for high degrees of viscosity. 
The matter of the spheroid is in fact so stiff that there is not time in 12 hours or a 
day to raise more than a very small tide, whilst in a fortnight a considerable lagging 
tide is raised. 
For e=44° the fortnightly tide has risen to give its maximum effect ( i.e ., sin 4e"= 1), 
whilst the effects of the other tides only remain evident in the hump in the middle of 
the curve. Between e=44° and 45° the ordinates of the curve diminish rapidly and 
the hump is smoothed down, so that when e = 45° the curve is reduced to the 
horizontal axis. 
By the theory of the preceding paper,* the values of e when divided by 15 give 
the corresponding retardation of the bodily semi-diurnal tide— e.g., when 30 the 
tide is two hours late. Also the height of the tide is cos 2e of the height of the 
equilibrium tide of a perfectly fluid spheroid— e.g,, when e=30° the height of tide is 
reduced by one-lialf. In the tables given in Part I., Section 7, of the preceding paper, 
will be found approximate values of the viscosity corresponding to each value of e. 
The numerical work necessary to draw these figures was done by means of Crelle’s 
multiplication table, and as to fig. 2 in duplicate mechanically with a sector; the ordi¬ 
nates were thus only determined with sufficient accuracy to draw a fairly good figure. 
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For the two figures I found 108 values of each of the seven terms of — (nine values 
& dt v 
of i and twelve of e), and from the seven tables thus formed, the values corresponding 
to each ordinate of each member of the family were selected and added together. 
From this figure several remarkable propositions may be deduced. When the 
ordinates are positive, it shows that the obliquity tends to increase, and when 
negative to diminish. Whenever, then, any curve cuts the horizontal axis there is a 
position of dynamical equilibrium ; but when the curve passes from above to below, it 
* “ On the Bodily Tides of Viscous and Semi-elastic Spheroids,” &c., Phil. Trans., 1879, Part I. 
