66 A Methed for ascertaining Latitude and Time 
sin second side sin angle comprised 
ae ual cides GEA (formula E): thus cosine 
a4 = cosine of angle opposite to second side = sine 
eee 5 a 
sine 3d side ? 
cording to the general formula (D). 
cos 6 sin h — sin ¢ cosh cos u 
Or eises Gee ete Gites ¥( 0) 
cos ¢ 
Such in fact is the formula (6) of M. Gauss, who finds . 
it, as well as formula (5), by combining the equations (1) 
and (3). ‘ = 
The formulas (3); (4), (5) and (6), contain, as we per- 
ceive, three unknown quantities instead of two; but they 
will soon serve to fiud the useful equations. 
By developing the formula (2), we shall have sin h’ = 
sin & sin g + cos é cos & cos g cos A + sin 4 cos # cos ¢ 
sin A. 
Let us substitute in this formula the values of sin ¢ 
(equation 5), of cos ¢ cos A (equation 6), cos ¢ sin A (equa- 
tion 3); we shall have sin h’—sin A sin 3 sin &” —sin A cos 6 
eos §.cos & — cos u cosh cos? sin ” + cos u cos h cos § 
sin dcos ¥ —sinucoshsin§cosd =0......F. 
Or, sin A’ —sin hf sin d sin & — sin A cos § cos dcos ” — 
cos uw cos h (cos 3 cos & — cos 6 sin &cos &) — sin u cos h 
(sin 6 cos o’) = 0. 
Or, sin h’—sin h sin 6 sin & —sin hk cos § cos } cos 
(cos 3 sin 5’—cos 6 sin cos 3) 
#” —cos Asin 6 cos & (>> tata peg ee cos u + 
sin u) = 0. 
Suppose, for the sake of shortening, 
(cos 3 sin & — cos § sin 3 cos 6’) 
mere aT e Mea e e cotang V ....(7). 
The equation will be reduced to sin A’—sin h sin 3 sin & 
— sin A cos 6 cos cos &” —cos h’ sin 8 cos® (cot V cos u + 
apni)" = 0; 
Or, sin /’ — sin Asin 3 sin & — sin h cos 6 cos é cos o” — 
, . '  fcos V cosu + sin V sin x 
cos fh sin § cos & ( ———_-—_.——__J= 0. 
sin V 
Or, sin h’ — sin f sind sin ” — sin h cos§ cos d cos” — 
cos h sin § cos 3 cos (V —u) 
(V) is known by means of the equation (7), therefore the 
last equation contains only the unknown quantity (V —~) 
or (u — V), for it may be the one as well as the other: we 
; shall 
