112 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
Section 16. 
Numerical calculation of the long inequality in the longitude of perihelion. 
„ d , . v! a! d R . ... . 
59. The expression tor being — . jj, the part which we have to 
consider may be put under the form 
d-zz' n'a! f x k /r k , , T 'a t x 
-jj = — ^ 2 . < 5 L . e 3 . cos (5 ® ) + 4 L . e 4 e . cos (4® + to) 
(10) (11) 
-f- 3 L . e ' 3 e 2 . cos (3 ^ -f 2 -f 2 L . e' 2 e 3 . cos (2 to' -f- 3 to) 
(12) , (9) 
+ L . e‘e* . cos (to -j- 4 to) + 3 M . e'^f 2 . cos (3 to' + 2 $) 
(10) (11) 
+ 2 M . e 2 ef 2 . cos ( 2 to' + w + 2 ^) -f- M . e' e 2 /* 2 . cos (to'-|- 2 to-J -2 6) 
+ N ( \ e f A . cos (to-' -f- 4 d) ^ cos 13 (n t + s') — 8 (n t + s) 
rid C ( 8 ) , ( 9 ) 
— ^ 7-^75 < 5 L . e 5 . sin (5 w') + 4 L . e ' 4 e . sin (4 to' -f ■&) 
(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 .ee*. sin (to' -J- 4 to) -j- 3 M . e' 3 f 2 sin (3 to' + 2 0 ) 
(i°) (11) 
+ 2 M . e 2 ef 2 . sin (2 to'to- f -2 6) -f-M .e’e 2 f 2 . sin (to'- f 2 to- f- 20 ) 
+ N ( e '/ 4 . sin (to' + 4 0 ) j- sin j 13 (n t + s') — 8 (n t + z) J 
which (neglecting the variable terms) is found to equal 
n X (92,35866) . cos {13 (n t+ s') - 8 (n t + s)} 
+ »' X (92,60190) . sin {13 (ra'£ -f- s') — 8 (nt -f s)} 
Integrating, 
to' = n' - (94,73673) . sin {13 (nt + s') - 8 (n t + s)} 
+ (94,97997) . cos {13 ( n ' t + s') — 8 (w t + s)} 
or 
to' = fl' + l",1250 . sin {8 (n t + s) — 13 (n f t -j-s')} 
