748 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
In the case where the viscosity is small this becomes 
d_l 
Tcdt 
9 
sin 4f[cos —X] 
(74) 
It will now be shown that the preceding result (71) for clj/dt may be obtained by 
means of the principle of conservation of moment of momentum, and by the use of the. 
results of a previous paper. 
It is easily shown that the moment of momentum of orbital motion of the moon and 
earth round their common centre of inertia is C£/Jc, and the moment of momentum of 
the earth’s rotation is clearly C n. Also j and i are the inclinations of the two axes of 
moment of momentum to the axis of resultant moment of momentum of the system. 
Hence 
£ • • 
— si nj=n sin ^ 
tv 
By differentiation of which 
P di . dn . . , .di 1 d% . . 
I it c °v= ( v sm l + na 0 Sl it-hM S11 U 
dn 
dt 
sin cos 
di 
dt 
COS J — 
'dn .. .. . .. . di 1 dlj_ 
-cos (*+.,)-»am 
sm j 
Now from equation (52) of the paper on “Precession,” the second term on the right- 
hand side is zero, and therefore 
k dt 
(*'+!)+ n cos ( ; +i) 
di 
dt 
But by equations (21) and (16) and (29) of the paper on “Precession” (when zs and 
k are written for the p, q of that paper) 
/]on 7-2 
— =-[-|cr 8 sin 4f 1 +2ra- 4 /< 4 sin 4f+^/c 8 sin 4fo + ur 6 /c sin 2gq 
ff nd k~ (A' — k~) ~ sin 2g+77r'A G sin 2g 3 ] 
n ^ = sin 4f x — ct 3 k 3 (ot 2 —k 2 ) sin 4f— \tsk‘ sin 4f, -j—^ hs- 5 /c(ot 3 -fi 3 k~) sin 2gq 
— ^nsK(nx~— K 2 ) 3 sin 2g(3zrr 2 + k 2 ) sin 2g s —| ct 3 /c 3 sin 4h] 
Then if we multiply the former of these by sin or 2 ztk, and the latter by 
cos or ct 2 — k 2 , and add, we get the equation (71), which has already been 
established by the method of the disturbing function. 
It seemed well to give this method, because it confirms the accuracy of the two long 
analytical investigations in the paper on “ Precession ” and in the present one. 
