DEPARTMENT OF TERRESTRIAL MAGNETISM. 315 



included by the sides h and c is the azimuth Z. We shall consider two cases: 

 (1) GiwenL, d,t, to ^ndh. (2) Given L, d, /i, to find L The case, given L, d, A 

 to find Z, may be derived from case 2 by the interchange of d and h throughout. 

 Formula (1) becomes, then, in navigation 



sin h = sm L sin d+cos L cos d cos t (2) 



Substituting in (2) 



2 sin L sin d = cos (L — d)—co& (L-^d) = a — j3 



2 cos L cos d = cos (L— d)+cos {L-\-d) = a+^ 



2sin/i = 27 



where a = cos (L — d), /3 = cos (L+d), 7 = sin h, we get 



27=(a-/3) + (a+^) cosf; or cos < = ^~'',"t^^ (3) 



Formulae (3) are convenient for use with the ordinary tables of natural 

 sines and cosines when a computing machine is available for making the one 

 multiplication in case 1 or the one division in case 2. 



For use in navigation, where the sides of the spherical triangle usually are 

 large, auxihary tables have been devised containing certain factors of sines 

 and cosines, thus greatly facilitating the multiplication or division. Multi- 

 plying (3) by a factor ft so chosen that 



/»(a+/3) = l+5 

 where S is small, either positive or negative, say numerically less than 0.1 

 we have 



2/i7=/ia-/,/3+(l+5) cos t; or, cos ^ = /^^-/^+^/<^ (4) 



Four factors are sufficient for observation of the Sun in latitude 60° or less. 

 At present two tables are prepared for observation of the Sun in latitude less 

 than 45' ; namely, /i = 0.56804 and fi = 0.68404. 



Each /j table is in two parts. Part 1 contains /< cos 6, which gives /<a for 

 6 = L — d, and/i/S for d = L-\-d, and part 2 contains 2 f^ sin h. In order to get 

 S directly by the addition of /<a and fS, 0.5 has been subtracted from each 

 value in part 1. Setting /^a —/</3 = 72 and 2/, sin h = H our formulae become 



H = R+{l+S) cos t; or, cos t^^j^ = {H-R)il-\-S') (5) 



where 1+*S' is the reciprocal of 1+»S. A small table gives AS, the numerical 

 difference between ;S and *S' for argument S; S and *S' having, of course, oppo- 

 site signs. 



In most cases S need not exceed 3 digits, so that the multiplication can be 

 most easily performed by the author's Interpolation Tables, Publication 245 

 of the Carnegie Institution of Washington. In the absence of any multiplica- 

 tion table, the product may be obtained by use of an auxihary table. Table 6, 

 based on the method of quarter-squares. 



Table 7 gives cos t for both arc and time. Al' the trigonometric tables are 

 to a minute of arc and to four decimal places, with the final figures so marked 

 that the even figure in the fifth place may be obtained when required. 



The following transformation of the formulae suggests another sUghtly 

 different form for the tables, using the versed-sine instead of the cosine. 



Substituting in (4) /<j;i = l+>S-/,a 



we have 2fh = 2/,a - (1 +5) (1 - cos i) ; or, vers / = ^-^'f"","^^ (6) 



or hav f=/,[cos (L-d)-sin h](l-{-S') 



In this form versed-sine t (or hav t) is always positive and only one of the 

 two parts of the fctahles is required. 



