THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
847 
terms of 22 different speeds, viz.: 9 in the first pair all involving 2 nt, 9 in the second 
pair all involving nt, and 4 in the last. 
Then since these five functions correspond to Diana’s tide-generating potential, 
therefore we are going to consider the effects of 22 different tides, nine being semi¬ 
diurnal, nine diurnal, and the last four may he conveniently called monthly, since 
their periods are -3 , I of a month and one month. 
We next have to form the £-§3-2? functions. We found that in the X-Y-Z 
functions there were terms of 22 different speeds ; hence we shall now have to 
introduce 44 symbols indicating the reduction in the height of tide below its equi¬ 
librium height, and the retardation of phase. The notation adopted is analogous to 
that used in the preceding problem, and the following schedule gives the symbols. 
Semi-diurnal tides. 
speed 
2n-m 
2n — 3fl 
2»-2/2 
2 11 — n 
2 n 2n+fl 
2n + 2J2 
2?i + 3/2 
2?i + 4/2 
height 
yiv 
piii 
F u 
F‘ 
F Fj 
Fai 
F iv 
lag 
2f iT 
2fiii 
2fii 
2fi 
2f 2fj 
24 
2fai 
24 
Diurnal tides. 
speed 
n — 4/2 
Cl 
CO 
1 
n- 212 
n — fl 
n n + fl 
91+2/2 
?i + 3/2 
71 “f" 4nf2 
height 
G iv 
G a 
G l 
G G; 
G u 
Gai 
Giv 
lag 
0 .iv 
0 .iii 
G-ii 
G 
O' 1 
O’ O’. 
tD Ol 
8ii 
gin 
§iv 
Monthly tides .* 
speed 
n 
2/2 
3/2 
4/2 
height 
H ; 
H m 
H iv 
lag 
D 
2h ii 
3h m 
4h iT 
The £-§3-2 functions might now be easily written out; for each term of the X-Y-Z 
functions is to be multiplied, according to its speed by the corresponding height, and 
the corresponding lag subtracted from the argument of the trigonometrical term. For 
example, the first term of £-—33- is F 1 E 1 P 4 cos (2y—/2— < 77 — 2f ] ). It will however be 
unnecessary to write out these long expressions. 
In order to form the disturbing function W, the £-§3-2; functions have now to be 
multiplied by the X'-Y'-Z' functions according to the formula (31). Now the X'-Y'-Z 7 
functions only differ from the X-Y-Z functions in the accentuation of /2 and ts, because 
Diana is to be ultimately identical with the moon. 
Then in the £-§3-2? functions fl is an abbreviation for flt-\-e, and in the X'-Y'-Z' 
functions /2' for /2^-fie'; hence wherever in the products we find /2 —/2', we may 
replace it by e—e'. 
* With periods of \, I, g and one month. 
5 Q 2 
