OF THE moon’s MEAN MOTION. 
401 
8. Also, assume 
/ 5 \ ^ 
alu~m'^{\—-e’^j cos {2v—2mv)-\-a^a^^&m {2v~2mv) 
3 2 / r I de' . , 
— pn e cos cTWj^+ajg^ sin c mv 
7 dd 
+^V cos {2v~2mv — c'mv)-{-a^^^^^\n {2v—2m)> — c'my) 
1 dd 
—-^rnde' cos {2v~2mv■^rc'mv)-\-a^^■^^sm {2y—2mv-\-c'mv), 
where the coefficients of the terms involving cosines are those given by the ordinary 
theory, and Ago, flig, and are numerical quantities to be determined. 
9. In developing the terms of the above equation, by the substitution of this value 
of the quantity ^ may be considered constant, and ^must be expressed in terms 
of it. 
dd ndt dd 
Thus Ty=-S ^ 
] 1 77 
■— cos {2y—2mv) — g mV cos {2v —2my — c' mv) 
d-^mV cos {2y—2mv-\-dmv)^. 
Also, integrating by parts, and putting 2 instead of 2 — 2m, 2 — 3m, and 2 — m in 
the divisors introduced by integration, since we only want to find the terms of the 
dd 
lowest order which are multiplied by we obtain 
sin {2v—2mv)-\-~e' sin {2v — 2mv — c'mv) 
— 2 ^' sin (2i' — 2mv-\-dmv) 
1 
/ 
3 5 , 2\ rrd , , , ^ 
= 2 y (^1 ~2 y (2 j' — 2mv)4-^ —e cos {2v—2mv—dmv) 
3 , 
— 4 ~a^ — 2my-\-c my) 
15 ndCj ddd ndt x 21 ndC dd ndt . , , 
+¥ vY'^t ■* <=“ (2»-2mv)-- cos (2»-2m.— cW) 
3 ndC dd ndt 
+4 Vy^^t (2.-2mv+cW). 
And aHv! = 3nde' sin c'mv\_—e' sin dmv] 
3 
2^ 
: — ^rrdd , 
retaining only the term which will be required. 
3 G 2 
