706 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
radius of terrestrial circle Z/ /c n +« b , 
radius of lunar circle L x a k x + (3 ' ~ ' 
According to the definitions adopted in (117) of /c 1 and k, 2 , is negative and 
(fc e +a)/a is positive ; hence L 1 has the same sign as L{, and L 2 has the opposite sign 
from L 2 . When t— — (mj- m^)/^- k z ), we have 
x=0, y=( — L. 2 )—L l , f=0, r)—L 2 -\-L{ 
Fig. 6. 
In fig. 6 let Ox, O y be the rotating axes, which revolve with a negative rotation 
equal to k. 2 , which is negative. Let M be the centre of the lunar circle, and Q of the 
terrestrial circle. Then we see that L and P must be simultaneous positions of the 
two poles, which revolve round their respective circles with an angular velocity k. 2 — k 1} 
in the direction of the arrows. 
M and Q are the poles of tivo planes, which may he appropriately ccdled the proper 
planes of the moon and the earth. These proper planes are inclined at a constant 
angle to one another and to the ecliptic, and have a common node on the ecliptic, and a 
uniform sloiv negative precession relatively to the ecliptic. 
The lunar orbit and the equator are inclined at constant angles to the lunar and 
terrestrial proper planes respectively, and the nodes of the orbit, and, of the equator 
regrede uniformly on the respective proper planes. 
In the ‘ Mecanique Celeste ’ (livre vii., chap. 2, sec. 20) Laplace refers to the 
proper plane of the lunar orbit, but the corresponding inequality of the earth is 
ordinarily referred to as the 19-yearly nutation. It will be proved later, that the 
above results are identical with those ordinarily given. 
Suppose then that 
