368 
MR. LUBBOCK’S RESEARCHES 
+ { - 2-8843819 r 19 + 3*7558426 R 12 + 2*3762929 Rj} sin (2 t-x- z) 
+ {2-16521 19 r 13 - 1-8430088 R 13 - 9-9454546 R l3 '} sin (2 t + * + z) 
+ { - 3-3350850 r 14 + 4-0300110 i? 14 + 2*6232753 JR 14 '} sin (« - z) 
+ { - 3-4377718 r 15 + 4-1169189 R n + 2-7021571 R lb '} sin (2 t - x + z) 
+ {2-1945476r l6 - 1-9298049 R l6 - 0-0433913 R ie '} sin (2 1 + x - z) 
+ {- 1-2465621 r 17 + 3*1977141 R 17 + 3-1977141 R 17 'j sin 2 2 
+ {2-0153626 r 18 - 2-1927325 R ia - 0*4983470 RJ} sin (2 t- 2 2 ) 
+ { 1 -8857018 r ls - 1-8857018 R w - 0-1575509 R l9 '} sin (2 t + 2 2 ) 
+ {- 6-4194035 r 101 + 7-1997263 B 101 + 5*69563 76 sin t 
+ { 3-1 744332 r 102 - 5-9693425 i? 102 + 5-8838691 R 102 '} sin (t - x) 
+ {4-2060990r 1O3 — 4*3466666 R i03 - 2-5333440 R 103 '} sin ( t + x) 
+ {- 4-3240929 r 104 + 5-1933516 R 10i + 37101653 R 104 '} sin (f - 2 ) 
[2-4899904] 
[1*3488787] 
[2*3541741] 
[2-3383041] 
[1-3946097] 
[3*4147879] 
[1*3033627] 
[1-1626061] 
[5*3481901] 
[6-4098870] 
[3-4884264] 
[3*6803018] 
The preceding expressions serve to show the extent to which the approxima- 
tion must be carried in the calculation of the quantities V, R, &c. 
If we take the term 5*6361652 R 3 , since log. ^- 3 = 77464329, it is evident 
that in order not to neglect * 01 " in the value of the coefficient of 
v ^~ cos (2 t—x) in the development of & R must be calculated exactly to the 
fifth place of decimals, but not beyond. The number 4-1857212 is the loga- 
rithm of the quantity , expressed in sexagesimal seconds, and 
serves to show in like manner how far the approximation must be carried in 
d R 
the calculation of 77 . 
d A 
When the square of the disturbing force is neglected, 
R = m ' fl3 
2 [x a 3 
m. a 3 
jfl o "" 
8 [x a, 3 
m { a 3 
2 [x a 3 
c != l+3r„-ii!!+=l 
ix a 3 
7 m ! a 3 
2 [X af 
The equation of p. 5, line 8 , gives r 8 = 0 . 
d aW4’-i/^ d '+^{/d>r 
t\ 2 — 3 r 0 r 8 — 3 Tfj To — 0 
