XXVI THE THEORY OF THE LONG INEQUALITY 



- , 4mn'd 2 P 2 ' . „ 4»m'a' 8 P 2 



Sa = - sin 2\ + cos 2X, 



2 



, , f wm'tani*/ d dP 2 Zmri tan ^ yd P, 



\ w sin l" \/l - e' 2 sin 7' d y "> sm l " 



■v/l -e' 2 sin 7 dy 



rnride' tan IV e?P/\ . 



£-£- — i sm2\ 



B 



2<o sin 1 de 



- (same with P 2 for P 2 ') cos 2\, 



»„, mn'd dP 2 ' . mn'd dP 2 



ell = — . — r sm 2X + . — , cos 2X. 



8o) sin l" v'l - e' 2 sin 2 l 7 ' dn 8(u s in l" \/l - e' 2 sin 2 l 7 ' dn 



29. log aP 2 = 3-5412437 -, log aP 2 ' = 4-0623939, 



3»i'w s aP 2 „ _ 3m'n 2 aP 2 



.-. — r-3 1 - - 132"-9360, — — £- = + 4"-4136, 



2«, 2 sm l" 2ft, 2 sm l" 



3mn' 2 a'P., ~ 3mri 2 dP 2 ' 



2 . ., 9# -9782, . „ = + 8**198, 



cu 2 sin 1 ft) 2 sm 1 



, t dP * - , * dp * - 



log a 8 — — = 2-2149551 -, log a 2 — — = 3-258721 1, 



da da 



m'na 2 dP 2 „ „ m'nd 2 dPl 



= + 8"-6791, -rZ~r> IT-- °- Q 599, 



o) sin l" da w sin l" da 



da a da da a da ' 



rnn'd* dP„ ,, . mn'd 2 dP 2 



... ___ — ? = - /'-5176, — — ; ~=+ 0"-7297, 



to sm 1 da to sin 1 da 



dP 



ae —-— = 2aiV, e 2 cos 2-ar + aN 3 ee cos (tst + •ht') 

 de 



dP 2 ' . , . 



ae = same with sines for cosines, 



de 



dP 2 - , dPl - 



.'. log ae — - = 37589679 -, log ae -~ = 3-0914348; 



de de 



m'nae dP 2 _ + ^^ _«W dP/ = _ ^^ 



4ft> sin l" de 4a> sin l" de 



a'e' — r = - J2aiV 2 e' 2 cos 2^+ aiV 3 ee' cos (tst + •sr')}, 

 de a 



dP' 

 de — — = same with sines for cosines, 

 de 



mride dP« „ mride dP 2 „ 



. „ — 7 = + 0"-1025, : — jf ~ = + 0"-1089 ; 



4(o sin 1 de 4oj sin 1 de 



.'. $(nt + e) = - 125" -016 sin 2\ + s"-6l7 cos 2\, 



8 (n't + e) = + 89"-358 sin 2X - 2"-599 cos 2\, 



