THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
829 
the equator and the orbit to their proper planes will continue to diminish, as the square 
root of the moon’s distance diminishes, and at an increasing rate. 
Suppose that, in continuing the integration, the solar influence be entirely neglected, 
and the motion referred to the invariable plane of the system. This plane will be in 
some position intermediate between the two proper planes, but a little nearer to the 
earth’s plane, and will therefore be inclined to the ecliptic at about 11° 45'. 
The equations of motion are now those of § 10, Part II., which may be written 
- G (' : +7) 
knf= D (i+j) 
But since i/jz=g]kn= 1/ltl, they become 
Jen log tan \j—— ~*~~^ m G 
hi jr log tan \ l— (1 + ttt)D 
*' g 
(compare with the first of equations (255) given in Part III., when t= 0 ). 
These equations are not independent of one another, because of the relationship 
which must always subsist between i and j. 
Then substituting from (263) (in which r' is put zero, and G, D written for T, A) 
we have for the case of large viscosity 
hi— log tan ^ j— — o(l +ttt) 
dg 
4A(1 — A) 
1-2A 
hi f^ log tan \i- 
i(i+rn) 
1 
4A(1 — A) 
1 —2A 
When A=^, 4A(1 — A)/(l — 2A) is infinite, and therefore both djfd£ and di/d£ are 
infinite. This result is physically absurd. 
The absurdity enters by supposing that an infinitely slow tide (viz.: that of speed 
n— 2/2) can lag in such a way as to have its angle of lagging nearly equal to 90°. 
The correct physical hypothesis, for values of A nearly equal to \, is to suppose the 
lag small for the tide n — 2fl, but large for the other tides. Hence when A is nearly 
= g, we ought to put 
„ l) . 2n , . 2(n — 2fl) 
sm 4L=— sm2g=-, butsm2g 1 = - 
1 n — fL & n oi p 
5 O 
MDCCCLXXX. 
