740 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
The seven gothic-letter functions defined by (41-4-7, 50-3-6-9) are functions of the 
sines and cosines of half the obliquity and of half the inclination, but they are 
reducible to forms which may be expressed in the following manner :— 
^ cos/ [l —^ sin 2 /— 2 sin 2 7(1 —■§ sin 2 /)-j--f sin 4 7(1 —\ sin 2 /)] 
d?i = i cos 7 [1 — I sin 2 /— I sin 2 7(1 — f sin 2 /)] 
(!^ 1 + ®o= —;|r cos,/[1 — sin 2 / —sin 2 7(1 sin 2 /)-f -§ sin 4 7(1 sin 2 /)] 
©q — C*Br 3 = —^ cos 7 [1 —f sin 2 /— 3 sin 2 7(1—f sin 2 /)] 
Jjr = ^ cos 7 \_\ sin 2 /-j- sin 2 7—f sin 2 7 sin 2 /] 
© = ^ cos 7 [l — sin 2 /— 2 sin 2 7-f-f sin 2 7 sin 2 /] 
$} = —i cos/ [| sin 2 /-f-f sin 2 7(1 —| sin 2 /) —^ sin 4 7(1—f sin 2 /)] 
> . (62) 
-J 
These coefficients will be applicable whatever theory of tides be used, and no 
approximation, as regards either the obliquity or inclination, has been used in obtaining 
them. 
7. Application to the case where the planet is viscous. 
If the planet or earth be viscous with a coefficient of viscosity v, then according to the 
theory of viscous tides, when inertia is neglected, the tangent of the phase-retardation 
or lag of any tide is equal to 19u/2 gciw multiplied by the speed of that tide ; and the 
height of tide is equal to the equilibrium tide of a perfectly fluid spheroid multiplied 
by the cosine of the lag. If therefore we put ^y~ = p, we h ave 
tan 2f x =-———, tan 2f=—, tan 2£ = 3(re+ - 1 ^ 
1 P P P 
n — 212 n n + 2fl , 2/2 
tan g : = —-—, tan g=-, tan 2 g 2 =—-—, tan 2n=— 
F : = cos 2f 1} F= cos 2f, F e = cos 2f 2 , G x = cos gq, G= cos g, G 3 = cos g 3 
and H= cos 2h. 
Therefore 
f = g {^i sin sIn 4f -d? 2 sin 4 4+<^i S4n 2g! 
+ © sin 2g—© 3 sin 2g 3 —sin 4h] . . . (63) 
This equation involves such complex functions of 7 and/, that it does not present to 
the mind any physical meaning. It will accordingly be illustrated graphically. 
For this purpose the case is taken when the planet rotates fifteen times as fast as the 
