XX THE THEORY OF THE LONG INEQUALITY 



, , mn - 



21. Since log a = 1*2829093, and log n = 1-6256531 and log = 3-4090935-, we have, 



CO 



2m'na?P „„ 2m'«a s P 



= + 0-003720466, = - 0-000560276, 



to co 



. ZmtfaP „ ■ 3m'n 2 aP' 



and = + 0'-012286620, = - 0-001850278 ; 



co w 



t 



and since loga'= 1*4776479, log»'= 1-3335464, and log — = 5*0684631 -, we have 



co 



4mn'a'*Q B „ 4mw'a' 2 Q' 



— = + 0-008369576, — = - 0-001207680, 



co co 



6mri 2 a'Q .. „ Gmn^a'Qt ,, 

 ■ + o -009009346, = - 0-001299995. 



CO 10 



Therefore, 



8n = + 0"*O01850278 sin \ + o"' 012286620 cos \, 

 $a = - -000560276 sin X - -003720466 cos X, 

 in'm - 0"-001299995 sin X - o"'009009346 cos X, 

 eSa'= +0 -001207680 sin X + -008369576 cos X. 



22. Since log tan \y = 2*11894, we have 



m'nae tan A y dP ,, m'nae tan A y dP" „ 



— y-t- TT = + ° 2511 > ~ i-T^ 2 -37 " " ° ° 525 ' 



w sin 1 de to sin 1 de 



m'na tan A -y _ i _ m'na tan i -v „, „ 



: — %1 P = + 0"-2630, j — iZ p'= _ o"*0396, 



co sin 1 co sin 1 



. mn'a'e tan i. -y' dQ ' mn'a'e' tan 1 -y' dQ' „ 



nd . — JLJ- -5 - + 0"*0095, . — ^_L -TL = + o"*0103, 



to sin 1 de W sin 1 de 



2mrid tan i y „ 2mria tan ^ y . „ 



. ,? Y Q = + 0"*3779, . J r #•= - 0"*0545; 



to sin 1 to sin 1 



.*. Sy ■ + 0"*092 sin X + 0"*514 cos X, 



Sn= o, 



tym - o"-065 sin X - 0"-368 cos X, 



«5n'= o. 



23. The part of B of the third order involving X is, (Art. 8) 

 R=rn (Pj cos X - P/ sin X), 

 where aPi = a (M 3 e s + M t ee' 2 + M 5 e sin 2 \ y') cos sr, 



+ a (J/ 6 e' 3 + Jf,eV+ Jlf 8 e' sin 2 l 7') cos or'. 

 + aiI/ 9 eV cos (2-ar - -zir'), 

 + aM 10 ee' 2 cos (2*5r' - -cr), 

 + ailf n e sin 2 ^ 7' cos (2 EL' - -or), 

 + aM r e' sin 2 ^ 7' cos (2 II' - -ar'), 

 and tzP/= same with sines for cosines. 



