lii THE THEORY OF THE LONG INEQUALITY 



and the corresponding corrections of 8e, elw, will be 



Sfe = - 3"*ll cos \, eXw - + 3"-ll sin X, 



the corrections of le, e'Sw' are less than l". 



Also Aa = + 0-00064982, Aa'= - 0*00210709. 



Hence $(nt+ t ) =—z—. — T , {AaP sin\ + AaP cosXk 

 w sin 1 l 



i. a o / z> t dP\fAo Aa'\ 



■where AaP = {a? + s ! — - > r , 



[ da ) \ a a J 



, rv f rv „dP'\/Aa Aa'\ 



AaP = \aP + a? —\[ - ; 



[ da)\ a a J 



.', <$ (nt + e) = - l"'l6 sin X ; the corresponding term of % (n't + e) is less than l". 



58. Those terms of the variations of the elements which depend upon 2\ and 3X may 

 be corrected with sufficient accuracy by means of the following formulae: 



f- 0-02128 Ae- 0-00452 e Aw ) . 

 d (nt + e) = { . , . , > sin 2\ 



' 1-0-02175 Ae + 0-02323 e A-ar J 



J + 0-00452 Ae - 0-02128 e Aw 1 



+ {- 0-02323 Ae- 0-02175 e Aw) 



cos 2X 



f - 0-00226 Ae - 0-00053 e Aw 1 . 

 + { , r. ,>sin3X 



[- 0-00236 Ae + 0-00240 e Aw ) 



{+ 0-00053 Ae - 0-00226 e Aw 1 

 , ,> cos 3\. 

 - 0-00240 Ae - 0-00236 e A-ar J 



L'.", , f+ 0-01532 Ae + 0-00326 e Asr] . 

 $(rit+ e ) = { „ . , - , A ,}sin2X 



v ' \+ 0-01566 Ae- 0-01673 e Asr J 



f- 0-00326 Ae + 0-01532 e A-ar ) 

 _■_ J \ cos 2\ 



\+ 0-01673 Ae'+ 01566 e Aw) 



{+ 0-00165 Ae + 0-00039 e Aw 1 

 + 0-00172 Ae'- 0-00175 e'Aw'j 



f-0 



sin 3\ 



- 000039 Ae + 0-00165 e Aw 1 



. , ,. ,>cos3\, 



00175 Ae + 0-00172 e A-ar'j 



which are obtained by differentiating the formulae 



» , . (Sm'n i aP 2 m'nd* dPA . 



(Zm'r&aPl m'na? dP'„\ 



+ ~) cos2X, 



oo da J 



2o> 



//* il flf 3 //' iv "i jj 



6 (nt + e) = — -j — sin 3X + — - cos 3X, 



m'n-aP 3 . m'rfaP* 



and the similar expressions for n't + e with respect to e, w, e', and or'. 



When the values of Ae, eA 13 ', &c, given in the last Art. are substituted, we have 

 ^ (nt + e) = + 0"-047 sin 2X - 13" -3 12 cos2X, 

 - 0"-027 sin 3X - l"-415 cos 3X. 



