THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
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satellite revolves. Then the speeds of the seven tides are proportional to the 
following numbers : 28, 30, 32 (semi-diurnal); 13, 15, 17 (diurnal) ; and 2 (fortnightly). 
It would require a whole series of figures to illustrate the equation for all values of 
7 and j, and for all viscosities. The case is therefore taken where the inclination j of 
the orbit to the ecliptic is so small that we may neglect squares and higher powers of 
sin^’. Then the formulas (62) become 
— 2 sin 3 * + f sin 4 i) 
cos — f sin 2 7) 
(U 1 +CBi 3 = — i(l — \ sin 2 7+f sin 4 7) 
(Gq — (Gh — —j cos 7(1 — 3 sin 2 7) 
&=■% cos 7 sin 2 7, (Gr=^ cos 7(1 — 2 sin 2 7) 
$$= — f sin 2 7(1 —f sin 2 7) 
From these we may compute a series of values corresponding to 7=0°, 15°, 30°, 45°, 
60°, 75°, 90°. (I actually did compute them from the P, Q formulas.) 
I then took as five several standards of the viscosity of the planet, such viscosities 
as would make the lag f : of the slow semi-diurnal tide (of speed 2n —2/2) equal to 
10°, 20°, 30°, 40°, 44°. Then it is easy to compute tables giving the five corre¬ 
sponding values of each of the following, viz. : sin 4f 1; sin 4f, sin 4f a , sin 2g l5 sin 2g, 
sin 2g a , sin 4h. 
Then these numerical values were appropriately multiplied (with Crelle’s three 
figure table) by the sets of values before found for the fi^’s, (Gr’s, &c. 
From the sets of tables formed, the proper sets were selected and added up. The 
result was to have a series of numbers which were proportional to dj / sin jdt. 
Then the series corresponding to each degree of viscosity were set off in a curve, 
as shown in fig. 4. 
The ordinates, which are generally negative, represent dj/ sin jdt, and the abscissae 
correspond to 7, the obliquity of the planets equator to the ecliptic. 
This figure shows that the inclination j of the orbit will diminish, unless the obliquity 
be very large. 
It appears from the results of previous papers, that the satellite’s distance will 
increase as the time increases, unless the obliquity be very large, and if the obliquity be 
very large the mean distance decreases more rapidly for large than for small viscosity. 
This statement, taken in conjunction with our present figure, shows that in general the 
inclination will decrease as long as the mean distance increases, and vice versd. This 
is not, however, necessarily true for all speeds of rotation of the planet and revolution 
of the satellite. 
The most remarkable feature in these curves is that they show that, for moderate 
degrees of viscosity (f : less than 20°), the inclination j decreases most rapidly when 7 
mdccclxxx. 5 c 
