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J iiew Solution of that Case of Spherical Trigononutr^, in 



which it is proposed, from Two Sides and tJteir contamed 



Angle, to determine the Third Side, By Edward Sakg, 



i Teaobep of Mathematics. (Communicated by the Author.) 



All the solutions that have hitherto been given for this case, 

 require the use of some auxiliary angle. When the other two 

 angles, as well as the third side, are wanted, the ordinary form 

 of the calculation is quite sufficient, but when the third side on- 

 ly is wanted, it introduces unnecessary work. Several attempts 

 have been made to determine the third side, independently of a 

 knowledge of its adjacent angles, but in all the methods that 

 have yet been proposed, the calculation consists of two distinct 

 operations. 



While in search of an easy plan for clearing the lunar dis- 

 tance from the effects of parallax and refraction, a solution of 

 this case occurred to me, which enables toe to determine the 

 third side by a single simple operation ; and which renders even 

 the computation of th^ other angles through its means more 

 simple than the common one. 



The different solutions of this case are so well known, and 

 its importance so easily recognised, that it is needless for me to 

 show wherein the method which I am about to offer, differs 

 from the known ones, or wherein its superiority consists. 



« and ^ being the known sides, and c their included angle, 

 the common formula. for the cosine of the third side, is 



cos y = COS a. COS /3 + sin a. sin /3. COS c. 



Multiplying each side by 4, and converting the products of the 

 sines and cosines into sums and differences, we obtain 



f 2 COS (« — /S) + COS (a — /3 — c) + COS (a — /3 + <^) 1 

 4 COS y =: I ^. 2 COS (« + /3) — COS (« + /3 — C) — COS (a + /3 + C) I 



Or, putting ^ for the difference^ and r for the sum of the sides, 

 f 2 cxw ) -f cos (J — d) 4- cos (J + o) ) 



4 COS y = 1^ 2 ^^ r— COS (^ — O) — COS {f-^e)] 



By means of this formula, the third side y is at once deter- 

 mined, and, except when it is either near 0° or 180% with great 

 precision. 



