THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
817 
k x and have now come to differ a little from — a and — -fi, but still not much. 
With these values I find 
log—— + 10 = 8-76472, log ^- + 10 = 8-69606 
Substituting in the formulas 
-. . a sin 2 i 0 . , b sin 2/ 
o/ = h - - 7 ., 6 i = h ---~ 
J 4 k^-\-ol cos 2j /c 1 + /3cos2^ 
I find 
Bj= 57' 31", Bi= 22' 42" 
Thus the oscillation of the lunar orbit has increased from 13" to nearly a degree, 
and that of the equator from 12 " to 23'. 
It is clear therefore that we have carried out the integration by the method of 
Part II., as far back in retrospect as is proper, even for a speculative investigation 
like the present one. 
We shall here then make the transition to the method of Part III. 
Henceforth the formulas used regard the inclination and obliquity as small angles; 
the obliquity is still however so large that this is not very satisfactory. 
§ 19. Secular changes in the proper planes of the earth and moon inhere the viscosity 
is small. 
We now take up the integration, at the point where it stops in the last section, by 
the method of Part III. The viscosity is still supposed to be small, so that y, 8 , g, d 
(as defined in (251)) must be taken in place of T, A, G, D, which refer to any viscosity. 
The equations are ready for the application of the method of quadratures in (250), and 
the symbols are defined in (251-4). 
The method pursued is to assume a series of equidistant values of £ and then to 
compute all the functions (251-4), substitute them in (250), and combine the equi¬ 
distant values of the functions to be integrated by the rules of the calculus of finite 
differences. 
The preceding integration terminates where the day is 9 hrs. 55 m., and the moon’s 
sidereal period is 8'17 m.s. days. If the present tropical year be the unit of time, we 
have, at the beginning of the present integration log n Q — 3'7445 l, log/2 0 =2*44836, 
and log Jc -\-10 = 6'20990, Jc being sflg of (7). 
The first step is to compute a series of values of n/n 0 , by means of (254). As a fact, 
I had already computed n/n Q corresponding to £=1, *92, ‘84, •76 for the paper on 
“ Precession,” by means of a formula, which took account of the obliquity of the ecliptic ; 
and accordingly I computed nfn Q , by the same formula, for the values of ^='96, * 88 , *80, 
instead of doing the whole operation by means of (254). The difference between my 
results here used and those from (254) would be very small. 
