2* Prof. J. Thomson on the true and extended 



above the horizon, and drawing from it two arc?, b and c, of 

 great circles to the horizon. Then, if we regard these two, 

 and the angle which they form, as given, we shall have one 

 triangle, which will answer the conditions of the problem, 

 bounded by 6 and c, and the less arc of the horizon joining 

 their extremities ; while with the greater or remaining arc of 

 the horizon they will form another triangle, which will an- 

 swer equally. The latter triangle is evidently composed of 

 the former, together with the hemisphere below the horizon. 

 7. The remaining angles will be found by the formulae 



^ „ cot bsmc — cos A cos c 

 cot B = 



and cot C = 



sin A 

 cot c sin 6 — cos A cos b 



sin A 



From these, since cot Q = cot (tt-}- Q), we shall have values 

 for B and C of the forms B', C\ and tt + B', tt-^-O, which 

 evidently correspond to the triangles above mentioned. 



8. When a side a, and the adjacent angles B and C, are 



given, the remaining sides may be found by means of the 



formulae 



, cot B sin C + cos C cos a 

 cot b = 



and cot c = 



sm a 



cot C sin B + cos B cos a 



sin a 



These give values for b and c, of the forms b\ d, and it-\-b\ 

 ttH- c/; the first of which are the values adopted in the books 

 on trigonometry. The meaning of the others will appear if 

 we enunciate the problem thus: Two great circles being 

 drawn through B and C, the extremities of a given arc a of 

 a great circle, and making with it given angles ; it is required 

 to find the distances between the points B and C and the two 

 points of intersection of these circles. These circles will in- 

 tersect in a point A ; and if they be continued through that 

 point each to a distance equal to tt, they will again intersect 

 in a point A". C A and B A are evidently the arcs i'and (/; 

 while C A A" and BAA" are respectively tt + ^' and %'\-d. 

 The third angle comes out, as it ought, of the forms A' and 

 2 ST — A', the former being the angle at A, and the other the 

 one at A". 



9. The case in which there are given a side and the opposite 

 angle, and either another side or another angle, to resolve the 

 triangle, may be treated in a similar manner; but neither 

 case presents any difficulty, and they exhibit nothing remark- 



% 



