On Practical Geodesy. 25 



These expressions are rigorously true, and can be used in 



other investigations. 



We have also from the triangles IS,D^, IS^^D,,, — 



sin 8, ' sin D, sin 8,, sin D,, , ^ 



sm A = r^^ — = Tv — ('0 1/ 



cos J 2 cos -I 5 ^ ^ 



l^p° In the "Account of the Principal Triangulation of 

 Great Britain and Ireland," the following expressions are 

 given — 



A = e^ • sin 2 A, • cos- (l, + I,) 'i^ / v 



A = 6^ • sin 2 A,, • cos^ {I, + Z,) • J S ^'''f 



That this formula is erroneous is easily seen : for indepen- 

 dent of the oversight committed in assuming that sin 2 A^ 

 is equal to sin 2 A^^, we know that any expression repre- 

 senting A must vanish when the latitudes l^, l^^, are equal ; 

 and this is not the case with formulae (102). 



29. When the stations S,,, S^o, are mutually visible (not 

 more than 100 miles apart), it is evident that if from the 

 middle point of the arc v we conceive perpendicular arcs 

 drawn to the circles S,D,,, S,,D,, they wiU form two right 

 angled spherical triangles (having vertices at S^ and S^J, 

 which may be considered equals in all respects. It is 

 evident that two of the sides of either of these triangles are 

 equals to J v and J :$, and that the third side of either may 

 be regarded as equal to J a • 



From this relation connecting the angle between the 

 normals, the angle between the normal-chordal planes, and 

 the circular measure of the geodesic arc between the stations, 

 we have — 



cos J 1/ = cos J A • cos J S (103) 



sin J A = sin I V • sin O (104) 



tan "I" A = sin J S • tan fl (i 5) 



tan J 5 = tan J v • cos O (los) 



simple relations which wiU be found very useful in practical 

 work of trigonometrical surveys. 



30. The following expressions for the cosines, sines, and 

 tangents of the angles of depression of the chord are 

 rigorous with respect to any two stations on the earth's 

 spheroidal surface; and the easy methods by which they 

 have been deduced (from what has been akeady done) are 

 omitted, as they can present no difficulty to the reader. 



