and the Methods of calculating their Results. 34-7 



south of it, and t the time of the nearest conjunction counted 

 from the first meridian. 



The development of the influence of A • e 2 depends on the 

 differential quotients in relation to e* of the quantities r cos 0' 

 and r sin 0' which occur in the expression for v and u. But 

 these quantities are differently expressed by e % and the ob- 

 served latitude 0, according as 0' denotes the declination or 

 the latitude of the zenith. The formulas to be employed in 

 the two cases must, therefore, be separately developed, while 

 all the preceding ones equally apply to both cases. I begin 

 with the case of 0' denoting the declination. 



We have in this case, r cos $' — — - — —. — — ; r sin 0' = 



(1 — e") sin (b , d.r cos <p' , r 1 sin a' 2 d r sin <p' 



whence — - - = r cos *' 



V(l-e 2 sin^) d.e 2 T 2(l-e2)*» d.e* 



. . r 1 sin (b'~ r sin <p' . r sin <B> 



= r sin d>' . ■ , ~ - — or putting 3 = - — - , 



Y 2(1— e 2 )* 1— e 2 ^ &r l- e 2 » 



d.rcosp' . „ , d.r sin«' . „, . -, „ 



T."*^" =2^ 3 ^cos0'; — ^— = £/3 2 .rsin$'-/3. 

 The expression for u and 1? being these : 

 u = r cos <$>' . sin (/a — A) ; z> = r sin <p' cos D — 

 r cos 0' sin D cos (ft— A), 



we obtain - — - = A/3 3 , u and - — ■ = i /3 3 . w— /3 cosD ; 



and, next, the part dependent on Ae 2 = w sin w . Ae 3 x 



[£/3 2 (wcos(N + 4/) -vsin(N + 4/))+ /3 cos D sin (N + ^)"j 



That part of this expression which is multiplied into | 3 2 

 maybe written: P cos (N + iJ/) -Q sin (N+4') — (P— u) x 

 cos (N + \J/) + (Q — v) sin (N + vJ/) ; and we may substitute for 

 P— u and Q— v their equivalents m sin M-f- n sin N. T', and 

 m cosM + n cosN.T' by which we have (P— u) cos (N + vf/) — 

 (Q-v) x sin (N+4/) =>.siri (M - N - 4>) -nT sin<J/ =; 

 (substituting for T' its value [5] — 



m . (cosM— N— ^)\ . OTsin(M — N) , 



n cos $ ' cos "4 1 



If we here apply the above transformation of Pcos (N + vp) 

 — Q sin (N + 4>) the expression dependent on A • e e will 

 become : = 



— 00 sin 7r . A ^ \jk /3 3 [* cos \J/ 4- ~ (t - ^ — t) sin ty — k~\ — . 



cos D sin (N+\f/)~|. 



[To be continued.] 



<2 Y 2 LIII. Notes 



