PARTICULARLY APPLICABLE TO SOME GEODETICAL PROBLEMS. 133 



This is identical with the following geometrical problem : 



"From three given points, to draw straight lines to a fourth in 

 then- plane, so that the lines may have to each other given ratios." 



This problem may be easily constructed geometrically by the theo- 

 rem of Art. 8. I shall here however give an analytical solution. 



Let A, B, C the angles of the triangle ABC, be the given points. 



Fig. 5. No. 2. 



r \ 2? *•' 



'•-tt- 



and D the point to be found; and employing the notation, of Art. 10, 



Let the sides of the triangle be «, b, c, 



the opposite angles being A, B, C, 



the distances of D from the angles x, y, x, 



the angles which the sides subtend at Z) ... a, /3, 7. 



There are given a, b, c, and therefore A, B, C, and the ratios 

 a; : y, x : », 1/ : x, to find x, y, %. 



Let the sides of a triangle conjugute to ABC (Art. 14.) be 



a, b', c, 



the opposite angles A', B', C . 



The angles of the two triangles must satisfy the conditions, 



A+A' = a, B+B' = li, C+C = y, (Form. II. Art. 14). 



It has been found (Form. iv. Art. 15.) that 



a' b' a c' b' c 



X : y = — : J-, x : z = - : - , >/ : x = j- : - . 

 •" a b a c '' be 



