THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
773 
For the moon: (i.) When the tide-raiser is the moon. 
(ii.) When the tide-raiser is the sun. 
For the earth: (iii.) When the tide-raiser is the moon, and the disturber the moon. 
(iv.) When the tide-raiser is the sun, and the disturber the sun. 
(v.) When the tide-raiser is the moon, and the disturber the sun. 
(vi.) When the tide-raiser is the sun, and the disturber the moon. 
The sum of the values derived from the differentiations, according to these several 
hypotheses, will be the complete values to be used in the differential equations (13), 
(14) and (18) for dj/dt, dN/dt, di/dt, dxfj/'dt. 
A little preliminary consideration will show that the labour of making these 
differentiations may be considerably abridged. 
In the present case i and j are small, and the equations (110) which give the 
position of the two proper planes, and the inclinations of the orbit and equator 
thereto, become 
l sin l Tt=~ cos 
n Jt =Ttj ' Shl 
.d'\lr 
n sin i-~— — (rr+r'f)/-— rtj cos ( N—\)j) 
(133) 
We are now going to find certain additional terms, depending on frictional tides, to 
be added to these four equations. These terms will all involve r 3 , r' 2 , or tt' in their 
coefficients, and will therefore be small compared with those in (133). If these small 
terms are of the same types as the terms in (133), they may be dropped; because the 
only effect of them would be to produce a very small and negligeable alteration in the 
position of the two proper planes.* 
In consequence of this principle, we may entirely drop W 0 from our disturbing 
function, for W 0 only gives rise to a small permanent alteration of oblateness, and 
therefore can only slightly modify the positions of the proper planes. 
Analytically the same result may be obtained, by observing that W 0 in (132) has 
the same form as W in (105), when i and j are treated as small. 
* For example, we should find the following terms in -sin_y’ , viz. 
Jc dt 
cos (N— -f) sin 2 g — + \(j+i cos (N—f)) [sin 2 2f x —sin 2 g x —sin 2 g]r 2 
S : 
which may be all coupled up with those in the second of (133). 
If the viscosity be small, so that the angles of lagging are small, it will be found that all the terms of 
this kind vanish in all four equations, excepting the first of those just written down, viz.: — 
MDCCCLXXX. 5 G 
