6q4 vhilosophical transactions. [anno J 775. 



Whence z=ida X -^- h - X vers. ^ = cosec. t x cosin. a X cosin. i x vs. 



(3 + cot. T X vs. ^. 



This solution is useful to find the distance of the moon from a star at some 

 distance from the ecliptic; in which case it coincides with the rule given by the 

 Astronomer Royal, Phil. Trans. 1764, vol.54, and which, taking in the cor- 

 rection here given, cot. t x vs. C,, will always be exact to a second. It is also 

 of use to find the declination of a star, whose longitude and latitude and ob- 

 liquity of the ecliptic are known. 



Solut. 3. Let the angle b be small, and the two legs ab, bc, very unequal ; 

 then the side ac will be nearly ab — bc Fig. 10. Put this = x., and suppose 

 AC = X -|- ^, then cos. ac = k — kz — k X vs. ^ = ad -{- ad — kz — k x vs. 

 ^ = bAD 4- ad, whence z = "^ ~ "*" — - X vs. ^ = sin. S X sin, » x vs. 

 P X cosec. i — a. — cot. A X vs. ^. 



Ex.— Let AB = 94° 36' 58"^ q 



BC = 23 28 24 >a8 in the example to »ol. 2. 

 B = 24 54 24. J 

 AB— BC = 71 8 34 cosecant 0.02396 



Sine A B 9.99859 



Sine BC 9-60023 



Versed of b 8.96851 



Sum = ^ nearly 2" 14' 11" sine 8.59129 



The value of ^ being without the limits of tab. 4, in the tables requisite to be 

 used with the Nautical Almanack, the correction cot. x x vs. ^ must be com- 

 puted thus : 



Cot. X 9.533 



V. sin. ^. . . . 6.881 

 Sum = cor. o' 53*, sine 6.414, this subtracted from the first value of ^, leaves 

 ^ = 2° 13' 18", which added to ,J — «, gives the side ac = 73° 21' 52'''. This 

 solution will help to find the sun's altitude near noon. 



I have dwelt the longer on this problem because it is one that is very com- 

 monly required in astronomical calculations, and the operation by the rules of 

 spherical trigonometry, in this as well as the next, is rather troublesome. 



Prob. 5. — Supposing the same things given, to find either of the angles, as for 

 instance c opposite the side ab. Fig. 10. — We have cot. c =: cot. b X cos. bc — 

 sin. BC X cot. ab X cosec. b = . Let w be an angle whose cot. — 



B BD ' ° M 



= cot. (3 X sin. S —a X cosec. S =i — ~ * , and suppose c = /* -f- ^; then 



m z , TO. vs. C boo — Ad i,,-, .^ Ad— Bab , m 

 cot. c 5= ,-\ r- = . Whence z = m X h - X 



M M' ' M^ BD BD ' M 



VS. ^ = sin.'' ju, X sin. « X cot. S X tang. ^(3 -j- cot, (a X vs. ^. 



