xlii THE THEORY OF THE LONG INEQUALITY 



With the same value of $R 



na r ( d(&ff) , dRA m'*na 2 (dP* dP 2 \ 



another secular perturbation of the epoch. 



50. Now let 



. dR 2 , dR 2 , dR 2 . , dR 2 . , 

 6R = -- r - d,e + — - Si-ar + —j 6,e + — > c^-ar 

 ae dsr de aTT 



then 2»a 2 / 



dR x dR l dR x dR x 



+ —j- d 2 e + -j— d 8 w + -rr d 2 e + — — , d 2 sr , 



de d-BT de d-w 



d$R) 



da 



^m^rfi 



2m 2 n 2 a 

 w sm 1 



2mm'nn'a 2 d 

 a? sin l" 



w 2 sin 1 



mm'nna'a' 

 or sin 1 



(dP (PP* dP^ d*P[\ . 1 

 \de deda de deda) I 



/dPd 2 i>i_di* d^\ r 



\ de deda de deda) ) 



Ida fP L+ d$ tpp,' \ gin J 



\de' deda de' deda) [ 



> fdQ dPPi_ _ dtf d 2 /^ / 



\de' de'da de' deda) \ 



r) 9in x ( 



d 8 P dP, (tf'P dR' 



\deda de deda de 



( 



( d 2 P dPl _d 2 P dPz} ( 



\deda de deda de ) J 



I d°P dPz tPP dP 2 \ . 1 



\deda de' de'da de ) I 



f d 2 P dP 2 d?P dP 2 \ r 



\de'da de' de'da de' ) \ 



d?P 2 „dNi, dN 3 , 



a 2 = 2a-—— e cos 2-ar + a 2 — ■ - e cos Osr + sr ) ; 



deda da da 



••■ log a 2 ^-^=1'77175-, 

 deda 



cPP' 



lo g« 2 -r-r-= 1-17260, 

 deda 



, cPP,, dN 2 , , . diV 3 



a 2 77* 2a 2 — — e cos 2ar + a"* e cos ( , 5r + ^r') ; 



de da da da 



drP 

 .-. loga 2 -—-^ = 1-70082 -, 

 deda 



log a 2 -i—r = 0*83633 -, 

 5 deda 



cPP' 

 log a 2 3-rf = 1-68990 -, 

 deda 



tPP' _ 

 loga 2 ——= 1-65660, 

 deda 



