802 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Now if we subtract the first of equations (224) from the third we shall find this 
relation to be satisfied. Hence the first and third equations are equivalent to only a 
single one. 
Similarly it may be proved that the second and third equations are similarly 
related. 
To prove that the four equations (224) reduce to those of § 6, when the nodes revolve 
with uniform velocity. 
It appears from § 13 that when a and b are small compared with a — /3, the nodes 
revolve with approximate uniformity, and the nutations of the system are small. 
If this be the case, we have approximately 
k 1 =—a, Ko— — /3. 
It will appear later that («'—/3')/(a—/3), a'/a, b'/b are quantities of the same order 
of magnitude as y, g, 8 , d. 
Now L x — J, the inclination of the lunar orbit to its proper plane, and L. z '= I, the 
inclination of the earth’s proper plane to the ecliptic. 
Therefore, the first and last of equations (224) become 
1 dJ 
gb — da 
J dt 
y- 
u — J3 
1 rtl _ 
s-i 
gb-da 
I dt ~~ 
h a-0 
But since the nodes revolve uniformly, b/(a— /3) and a /(a —/3) are small, and there¬ 
fore the latter terms of these equations are negligeable compared with the former. 
Hence 
1 dJ 1 dl ~ 
J dt~~ 7, I dt~~ 
These results in no way depend on the assumption of the smallness of the viscosity 
of the planet, and therefore we may substitute F and A (see (1 74)) for y and 8 . 
A comparison of the expressions for F and A, with those given in Part II. for djjdt 
and in my previous paper for di/dt, will show that our present equations for clJjdt 
and di/dt are what the previous ones reduce to, when i and j are small. But this 
comparison shows more than this, for it shows that what the equation ( 61 ) § 6 really 
gives is the rate of change of the inclination of the lunar orbit to its proper plane, and 
that the equation ( 66 ) of the paper on “ Precession ” really gives the rate of change of 
the inclination of the earth’s proper plane (or mean equator) to the ecliptic. 
