112 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
Section 16. 
Numerical calculation of the long inequality in the longitude of perihelion. 
. d to', . n' a' d R , ... . 
59. The expression tor -jj being — . j-g, the part which we have to 
consider may be put under the form 
d to' n'a ! f T ( 8 ) . ,\ T W 'a (a • i \ 
-jj — — ^ 2 . j 5 L . e 5 . cos (5 to- ) + 4 L . e 4 e . cos (4 to -f- to) 
no) rn) 
+ 3 L . e' 3 e 2 . cos (3 to' -{- 2 to) + 2 L . e' 2 e 3 . cos (2 to' -f 3 to) 
+ L ( \ e' e 4 . cos (to' + 4 to) -f- 3 \ e^f 2 . cos (3 to' + 2 6) 
( 10 ) ( 11 ) 
+ 2 M . <?' 2 e/’ 2 . cos (2TO' + ^d-2^) + M . e' e 2 f 2 . cos (to'+2to+2 6) 
+ N ( \ e / 4 . cos (to' -f- 4 6) cos 1 13 {n t + s') — 8 (n t + s) j- 
7 / syJ C ( 8 ) (9) 
— ^5 < 5 L . e ' b . sin (5 to') -J- 4 L . e' 4 e . sin (4 to' + to) 
( 10 ) ( 11 ) 
+ 3 L . e' 3 e 2 . sin (3 to' -{- 2 to) + 2 L . e' 2 e 3 . sin (2 to' -f- 3 to) 
( 12 ) . (9) 
+ L . e e 4 . sin (to -f 4 to) -{- 3 M . e' 3 f 2 sin (3 to' + 2 0) 
(i°) (ii) 
-f 2M . e' 2 ef 2 . sin (2 to' to - j- 2 6) -{-M . e'e 2 f 2 . sin(TO-'-J- 2 to -{- '26) 
+ N ( \ e'/ 4 . sin (to' + 4 6) | sin 1 13 {n t + s) — 8 {n t + s) | 
which (neglecting the variable terms) is found to equal 
n X (92,35866) . cos { 13 («' t+ s) - 8 (» t + g) } 
+ n X (92,60190) . sin {13 {n't + s') —8 {nt + s)} 
Integrating, 
to' = IT - (94,73673) . sin { 13 {n t + s') - 8 {n t + s ) } 
+ (94,97997) . cos {13 {n't + s') - 8 {n t + s)} 
or 
to' = IT + 1",1250 . sin {8 {nt + s) — 13 {n' t +s')} 
