213] 
IN THE LUNAR THEORY. 
295 
these departures being measured on the moon’s orbit; and, in like manner, measured 
on the sun’s orbit from a departure point on this orbit, but called for distinction 
“ longitudes,” instead of departures, 
«■', the longitude of the sun’s pericentre, 
®, the longitude of the moon’s ascending node; 
then we have 
tj = ct — £, 
tj' — Vj' — % ] 
and the disturbing function i2, so far as it depends on the position of the moon, is 
a function of the seven elements a, e, g, hj, rj, 1, ®, and it contains also the quantities 
a, e', g\ which relate to the sun. 
V 
Proceeding now to develope 12 in ascending powers of >, we have 
where, however, the last term is neglected in the sequel, and since we are only con 
cerned with the differential coefficients of 12 in regard to the lunar elements, the 
first term m' \ which depends only on the solar elements may also be neglected; we 
have 
cos H = cos 2 |<b cos (/-/ + tj -t') 
+ sin 2 COS (/’+f' + tj + tj'), 
and thence 
cos 2 H = cos 4 
+ 2 cos 2 |d> sin 2 
+ sin 4 
cos 2 (/—/' + C —C') 
cos (f-f' + tj-V) cos (/+/' + tj+V) 
cos 2 (/+/' +fc+C'), 
cos 3 H — cos 3 £<f> 
+ 3 cos 4 |<f> sin 2 
+ 3 cos 2 sin 4 
+ sill 3 ^<f> 
cos 3 {f—f' + <0 — tj') 
cos 2 (/—/'+ tj — tj') cos (/ +/' + tj +1>') 
cos (/— /'+1> — tj') COS 2 (J +f -1- tj +tj') 
cos 3 (f+f' + tj+tj'), 
and converting the powers of the cosines of f-f' + tj-tj',f+f' + tj+tj' into multiple 
cosines, and expressing the coefficients in terms of r\ (= sin and neglecting r\, we 
have