746 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Sidereal diurnal term. 
clN' 
7m K 
(Qk-\-Pzv) = 
Therefore 
and 
^=1 
<f)(N, e)w 2 = - , <xtk(sx 2 — k 2 ) 2 
V ~ 1 
<£(iV } e) W 3 = tsk(tP — K~y~G sing’ 
PI J P / / r ' A 
and -rrzAzs trr — K k ) = 0 
cUV v — -' 
Fast diurnal term. 
By symmetry 
e) W 3 = <xtk 5 (3ct 2 — k 2 )G 3 sin g 3 
Fortnightly term. 
and 
cf>(N, e)w 0 : 
Whence 
d [ r A2 
~dN^^ Z 
-1 
q{i<Q — Prd) 
«2h 
2/-1 
O 0 
7&~K" 
«2h 
2t ttk(Qk — -f- 4 ct 2 k 2 ^ = ^7 =F y2ra , K(rrr 2 — K 2 ) 
4>(F, e) W 0 =3<T7 2 k 3 (ttt 2 — k 2 )H sin 2h 
Then collecting terms we have, on applying the result to the case of viscosity, 
— q ^'=— [i|t«7 7 k sin 4f 1 d-7n- 3 K 3 sin 4f-f-^zrr/<: 7 sin 4f 3 -f-frrr 3 K 3 (rar 2 — k 2 ) sin 4h 
—■gu7 : v(T7T 2 —3 k 2 ) sin 2g 1 -)-)>73-K(CT 2 — k 2 ) 2 sin 2g-f-^CTK 5 (3ET 2 — k 2 ) sin 2g 3 ] . (71) 
In the particular case where the viscosity is small, this becomes 
— f ~= i— sin 4f to-k= 4 : 7 ^ sin 4f sin .(72) 
h dt 2 g 4 g v v ' 
The right hand side is necessarily positive, and therefore the inclination of the orbit 
to the invariable plane will always diminish with the time. 
The general equation (71) for any degree of viscosity is so complex as to present no 
idea to the mind, and it will accordingly be graphically illustrated. 
The case taken is where n/12=15, which is the same relation as in the previous 
graphical illustration of § 7. 
The general method of illustration is sufficiently explained in that section. 
Fig. 5 illustrates the various values which dj/dt (the rate of increase of inclination 
to the invariable plane) is capable of assuming for various viscosities of the planet, and 
