116 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
Section 20. 
Numerical calculation of the perturbation in latitude. 
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65. The first of the terms found in the last section is — M . e' 3 ff. cos 
{13 (n't + s') — 8 {nt + s) — 3 w' — d — 0}. With respect to this term only, 
^ = - M (9) . e' 3 f. cos {13 (n t + s') - 8 (n t + s) - 3 d - 0- 6} ; whence 
, rid pd R n 1 e‘ z f , 
6 = 0 Z? = 0 I3ri-8n -_-fsm{13(nt + t) 
— 8 (nt z) — 3 ts — d — 6}. 
And 
\ e 3 f f . sin {13 (n! t + g) — 8 (n t -f* s) — 3 zs — d — 6} ; 
whence 
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/ n ' a ' R ., . M a 
?=<*>' + piJ7r = v+ 
f'fVi </S' — - ' ft' • 13 n’ - 8 is t ”/- COS {13 («' l + s') 
— 8 (« £ + s) — 3 sr' — d — 0}. 
The Earth’s latitude, neglecting small terms, is f . sin (n t -f- s' — d ). And 
from the expression above, sin (ri t + s' — d) — 
M (9) a! 
A.VJ. U ’if, 6^^ f 
sin (n't + s' - 0')- p- . ^ - ri _ s ?i • ~fr - cos («'* + * ~ 0/ ) • si n 
{13 (n't + s') — 8 (nt + s) - 3 w' — 4' — 
Multiplying this by the expression for <p', and putting d, <p', for 0', O', in the 
small terms, we find for the latitude 
O'. sin (n t + s'— 0') — 
M (9) a' 
y. ■ lHw'-Tn ■ e ' 3 f • sin {12 (n't + s') 
— 8 (n t + s) — 3 vs — 6} 
and the last part, or the perturbation in latitude, is 
~ 13V - 8 n * e V- ~~ . sin {12 (nt s') - 8 (n t + s) - 3 vs - 9} 
Similar expressions will be obtained from all the other terms. 
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