XVU1 THE THEORY OP THE LONG INEQUALITY 



,, , / mria dQ mria'e' 2 dQ\ . 



eimr = : — r -— + : — T ——, sin \ 



V w sin 1 de 2co sin 1 de I 



+ (same with Q' for Q) cos X, 



. , Gmn'^a'Q' . 6mn'*a'Q 



tin = H sin X — cos X, 



to to 



* , 4wjw' a' 2 Q' . 4»m'a' 2 Q 



da = sin X + cos X, 



a> to 



- , I Zmri a tan l«y' , mn'a'e tan A«y dQ'\ . 

 07 = ( .,_ .» ' - Q - - _ : _ ,/, - I sin X 



to sin l" vo sin l de' 



- (same with Q for Q') cos X, 



SIT = 0. 



17. By means of quantities given in (Arts. 13 and 15), we shall find 

 log aP = 2-5775645 - , log aP = 3*7553692 ; 



and therefore, by (Art. 15), 



3m'« 2 aP „ „ „ Srrin i aP f 



. ■ ,„ = - 2890"-6362, . - + 435' -3096. 



a) 2 sin 1 a> 2 sin 1 



aP 



. aP ( IN. 



Again, «Q = 2 a e cos -or , 



a V 4a 2 / 



, . aP ( 1 \ , . 



a \ 4a 2 / 



.-. log a'Q = 2-7745325 -, log a'Q' = 3'9337808, 



6mn' 2 a'Q ., . 6mri 2 a'Q' „ 



■ ■ ,„ = - 2119"-6020, . , = + 305 -8459. 



a/ sin 1 a) 2 sm 1 



18. By (Arts 14 and 15), 



AP - '.«*#- 



log a 2 = 1-0349352 -, log a 2 = 2-1300441, 



da da 



2m'na 2 dP .. „ 2w»'w« 2 dP" „ 



, . -T--+ H4"-6790, r— 7T -J- - ~ 14-2755, 



u> sin 1 da to sin 1 da 



, dQ 1 , . dP _. / n , 



and a 2 — 7 = — (a 2 — — + aP) + 4 a + — - e cos •#• , 



da a da ' \ 8a7 



„ *# 1 dP* / 1 \ , . , 



a 2 -—7- = (a 2 -y- + aP) + 4 a + — -) e sin tst , 



da a da \ 8a 2 / 



-. log a' 2 — - = T-4001832, log a' 2 - — r = 3-7675884 -, 



da da 



« 



2»2«'a' 2 dQ ,, . 2»rara'a' 2 dQ' ,, 



: . ., 3-7 = - 12l"-3672, . jj -SL m + 2"-8282. 



a) sm 1 aa a> sm 1 da 



