90 PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
29. The terms collected in (20), (24), and (28), form the complete coefficient 
of cos (13 — 8) in the development of R to the fifth order. The arguments of 
the cosines multiplied by the different series are all different; so that there 
( 8 ) 
are twelve different terms to be calculated. Using the symbols L , &c., the 
complete term is expressed thus : 
( 8 ) 
L . e' 5 . cos {13 (n 1 1 -f- s') — 8 (n t -f- s) — 5 w'} 
(9) 
+ L . e' 4 e . cos {13 (n t + 0 — 8 (n t -f- s) — 4 & — w} 
(10) 
+ L . e' 3 e 2 . cos {13 (n't + s') -8 (nt + z) - 3®'- 2®} 
(11) 
+ L . e' 2 e 3 . cos {13 (n 1 1 -f- s') — 8(»(-f s)-2®'-3®} 
(12) 
+ L . e' e 4 . cos {13 (n 1 1 + s') — 8(«Hs)-®'-4®} 
(13) 
-j" L . e 3 . cos {13 (n* t -J - %') — 8 (n t -j - s) — 5 
( 9 ) , 
+ M . e^f 2 . cos {13 (n 't -f- 0 — 8 (n t + s) — 3 to-' — 2d} 
(10) 
+ M . e' 2 e f 2 . cos {13 (n! t -f- s') — 8 (n t + s) — 2 w 1 — ■& — 2 0} 
(11) 
+ M . e' e 2 f 2 . cos {13 (n! t -j- s') — 8 (n t + s) — zs — 2 sx — 2 
(12) 
+ M . & f 2 . cos {13 (w 't + s') — 8 (n t + s) — 3 — 2 0} 
(10) 
+ N . e'/* 4 . cos {13 (■«' t + s') — 8 (w t + s) — tzx' — 4 0} 
-{- \ ef x . cos {13 (n't + s') — 8 (n £ + s) — to- — 4 0} 
\ 
Section 11. 
Considerations on the numerical calculation of the inequalities in the Eartlis 
motion depending on this term. 
30. If we examine the expressions of (2), it will appear that the values of all 
may be deduced with little trouble from the terms above, except that depending 
on Since a' enters only into the coefficients, will be produced by 
