742 
MR. G. H. DARWIN ON THE SECULAR CHANGES US' 
the obliquity is zero; whilst for larger viscosities (f x between 20° and 45°), there is a 
very marked maximum rate of decrease for obliquities ranging from 30° to 40°. 
Diagram illustrating the rate of change of the inclination of a satellite’s orbit to a fixed plane on which 
di 
its nodes revolve, for various obliquities and viscosities of the 
planet (~— 
\amj 
dt 
when j is small 
> 
We now return to the analytical investigation. 
If the viscosity be sufficiently small to allow the phase retardations to be small, so 
that the lag of each tide is proportional to its speed, we may express the lags of all 
the tides in terms of that of the sidereal semi-diurnal tide, viz. : 2f. Then on this 
hypothesis we have 
And 
sin 4fj sin 4f sin 4f 2 sin 2gq 
A n- i Th- T y • , - 1 "1 A 5 • A <* - v> 
sin4f sm4f sin 41 sin 41 * 
sin 2g 
sin 4f 2 
sin 2g 3 
sin 4f 
JL 
2 
+ A 
sin 4h 
sin 4f 
= X, where \ = 
n 
n 
g *j 
k sin y dt 
£sin 4f[d? 1 +df-d? i +4(<!K 1 +<ffi-<!S^-X(dP 1 +df 3 -l-«S 1 +ffi i +?e)l 
23 
But by (62) 
\ (CBq—(Sxo)= •§• cos i and = ^ cos i 
and 
&1+&-2+&1 + + il) = 0. 
