36 On Practical Geodesy. 



However the differences are very small indeed. As an 

 instance we may consider the large spheroidal triangle 

 of article 7, page 234, of the "Account of the Principal 

 Triangulation of Great Britain and Ireland." Here we find 

 that at the station whose latitude is 53°^^ SO', the spheroidal 

 angle exceeds the corresponding angle of the Legendre sphe- 

 rical triangle by about y^^ of a second ; and, although such 

 may . be disregarded in actual practice, it is nevertheless 

 obvious that the usual method of manipulating the measured 

 angles of a spheroidal triangle (by means of Legendre's 

 theorem, so as to have their sum give the desired spherical 

 excess) is erroneous in principle. 



NOTES. 



It is easy to perceive that the principal theorems arrived 

 at apply to any surface whatever as well as to the surface 

 of the spheroidal earth, even when such surface is so irre- 

 gular as to be inexpressible by means of an equation. 



We can assume any straight line cutting the normals to 

 the surface at the stations S^, S^^, as polar axis of reference ; 

 and then, assuming any point C^ in this polar axis as centre 

 of reference, we can take the plane through it perpendicular 

 to the axis as the equatorial plane of reference. Thus the 

 figure can be constructed as already indicated in the case in 

 which the surface is a spheroid; and we have formulae (50), 

 &c. 



When the stations S^, S^^, are so near to each other as to 

 permit us to regard the normals as making angles with the 

 chord such that the ratio of their sines can be regarded as 

 equal to unity, and the traces of the normal-chordal planes 

 as equals in length and circular measure, we have — 



tan 1 CO = ^-^\4^A ' ^^* i (^^ + ^ J 

 sm i {I, + I,) 



tan 



. tan il" = - cos i (A/+ A, + a>) 

 cos i (A, + A,, — (o) 



sin A, R,, cos I, 



sin A„ R, cos l. 



and all the formulae not implicating peculiar properties of 

 the spheroid. If there be three stations to be simultaneously 

 considered, the assumable position for the polar axis of 

 reference is generally restricted, as such axis must cut the 

 three normals to the surface drawn through the stations. 



