IgO Ocean Magnetic Observations, 1905-16 



result is required within 0?02 only. If m is not zero, but less than 1°, then the arc can be 

 substituted for its sine, and its cosine can be taken as unity, and (2) may be written 



cos A = cos A sec h — m tan h 

 Assuming cos A' = cos A sec h, we get 



cos A' - cos A = - 2 sin 2 01' + A) ?>m-^{A' - A) = m tan h. 

 When A' - A is less than 2°, the arc may be substituted for its sine, and 



A' — A = —m tan h cosec ^ {A' + A) 



This equation will give A' — A by a series of rapid approximations, each one furnishing 

 a new and closer value of A'. 



If A and A are nearly equal, it will be expedient to determine the angle A in another 

 way, by computing and tabulating the small angle A - A, as follows: (2) may be written 



cos A = sin /i sin m + cos h cos m cos A 

 and we also have 



sin m = m 





+ etc. 



cos m = 1 — 2 24 " 



Let us assume 

 From (4) we obtain 



A = A - X 



(4) 



x» . 



or ju 



cos A = COS (A - x) = cos A + X sin A - -2 COS A - g- sin A + etc. 



Substituting the expressions for sin m, cos m, and cos A in the equation for cos A, we have 



cos A = 



wr 



+ m sin h — ^sin /i + etc. 

 b 



x^ x' 



+ cos h cos A + X cos /i sin A - -^ cos /i cos A - -g cos /i sin A + etc. 



x^m* 



--prcosh cos A ^— COS /i sin A + . 



2 4 4 



COS h COS A + etc. 



and from these we obtain the following general expression for x; 



X = 



m" 



— m tan h cosec A + -^-tan ^ cosec A - etc. 

 + cot A (sec /i - 1) + •2 cot A + g- - etc. 



, TO- , . , XTO" 

 + yCOt A+-^ 



etc. 



When weather conditions permit in actual work, the Sun is taken so low that A - A 

 rarely exceeds 1?5. Usually to should be even smaller; so, in general, the series for x is 

 rapidly convergent. Let m and x be expressed hereafter in degrees; then we have 



X = 



- TO tan h cosec A + 0.00005 7n^ tan h cosec A - etc. 

 + 57.3 cot A (sec /i - 1) + 0.00872 x- cot A + 0.00005 x^ - etc. 

 . + 0.00872 TO^ cot A + 0.00015 x nn^ - etc. 



(5) 



The two principal terms of this expression give an approximate value of x, which may be 

 used in calculating the subsequent terms, if desired. Ordinarily, this first approximation 

 suffices, and equation (4) then reduces to 



A = A - 57.3 cot A (sec h - \) -\- m tan h cosec A 



