32 U. S. COAST AND GEODETIC SURVEY. 



In equation (84) the functions involving I and x may be expressed 

 in the general form cos (2l + a) or cos a, in which « = ± 2%, ( ± x — W2) , 

 or zero. 



In equation (85) let 



lc = 2e sin {s — p)+ 5/4 e^ sin 2 (s — p) 

 + 15/4 me sin (s-2h + p) + n/8 'm? sin 2 (s-7t) (89) 



Then 



Z = (r + fc (90) 



The maximum value of li. is small. If numerical values for e and 

 m be substituted in (89) , it is found that the maximum value of 2k 

 is 0.273 of a radian, the sine of which is 0.270. It may therefore be 

 assumed without material error that the sine of any angle not greater 

 than 2k is equal to the angle itself. 



Then 



sin 2k = 2k = ^e sin (s — p)+5/2 e^ sin 2 (s — p) 



+ 15/2 me sin {s-2h + p) + 11/4: ni" sin 2 (s-li) (91) 



and 



cos 2fc = l-2 sin^ k=l-2¥ 



= 1-8 e^ sin2 {s - p) = 1 - 4.6^ + 4:6^ cos 2 (s-p) (92) 



if terms smaller than those of the second order are neglected. 

 From (90), (91), and (92) we may now obtain 



cos (2Z + Q!) = C0S {2a + 2k + a) 



= COS 2k cos (2o- + a) — sin 2k sin (2a + a) 



= [1 - 4e2 + 4e^ cos 2 (s - p)] cos {2a + a) 



— [4e sin (s — p)+ 5/2 e^ sin 2 (s — p) 



+ 15/2 me sin (s-2?i + p) + 11/4 m^ sin 2 (s-/i)] sin (2a + a) 



= (1-4:6^) COS (2(7 + a) 



+ 2e cos (2o- + a + s — 2>) — 2e cos {2a + a — s + p) 



+ 13/4 g2 cos {2a + a + 2s-2p)+3/4 e^ cos (2(r + a-2s + 22>) 



+ 15/4 me cos (2o- + a; + s — 2^ + 2?) — 15/4 me cos (2o- + q! — s + 2^ — 2?) 



+ 11/8 m^ cos (2(7 + a + 2s-2/i) - 11/8 m^ cos {2a + a-2s + 2h) (93) 



The general coefficient of (84) may be written 



3/2|| = 3/2|«;(0 (94) 



and the variable part of each of the terms in (84) may then be ex- 

 pressed by one of the following general forms : 



©^ (IJ 



cos a; or ( -7 j cos {2l + a) 



The value of the first is given in equation (88). For the second 

 we have from (88) 



( -7 j cos a = (1 + 3/2 6^) cos a 



+ 3/2 e cos {a + s — p) +3/2 e cos (a — s + p) 



+ 9/4 e^ cos (Q; + 2s-2p)+9/4 e^ cos (a-2s + 2p) 



+ 45/16 me cos {a + s — 2h + p) +45/16 me cos (a — s + 2^ — p) 



+ 3/2 m= cos (a + 2s - 2h) + 3/2 m^ cos (a - 2s + 27t) (95) 



