ofLongiUidefrom the Latitudes and Azimuths of two Stations. 27 



as a means of verification, and of judging of the exactness and 

 consistency of the data. These formulas may be of use, when 

 the excentricity is given. But if the excentricity is sought, we 

 must find/"; and the least attention to the formula ^ov f, will 

 show that practically no dependence can be placed on the re- 

 sult obtained by it, more especially when X — x' is very small : 

 because a small change in A — A', within the limits of the errors 

 of observation, as half a second, will produce a great change 

 in f, or will even make it vary from a certain value, to be 

 double of that value. 



It has been proposed to find the excentricity by what is 

 called the equation of the geodetical line. But this equation 

 is liable, even in a greater degree, to the objection of which 

 we have been speaking. It is in reality very pliant and accom- 

 modating; for let us take the data as actually observed, and 

 we will obtain by it a certain excentricity ; but we have only 

 to change the data a little, within the limits of the errors of 

 observation; and the same excentricity will become zero, or 

 any other quantity we please. It is asserted that, in the in- 

 stance of Beachy Head and Dunnose, the compression is yf^, 

 and that it can be no other. But if we calculate the length of 

 the chord between the two stations, with the compression men- 

 tioned, the known length of the radius of the earth's equator, 

 and the difference of longitude in the Survey, it will be found 

 to fall short of the measured chord no less than 80 fathoms ; 

 which is a proof, if any proof were needed, how little reliance 

 is to be placed on such computations. 



In the Survey no use is made of the azimuths, except to find 

 the difference of longitude. The degree perpendicular to the 

 meridian is'deduced from the length of the chord between the 

 stations. But without computing the perpendicular degree, 

 we may determine, directly from the diflference of longitude 

 and the chord, the dimensions of a spheroid that will repre- 

 sent the measurements both on the meridian and perpendicular 

 to it. For, the difference of longitude being given, the arc /3', 

 which is the base of the triangle on the surface of the sphere, 

 will be known ; and, if a be the radius of the earth's equator, 



and y the chord of /3', we shall have y' = 2a sin — -. Again, 



y being the measured chord on the earth's surface, if we form 

 the expressions of y'- and y- by means of the coordinates at 

 the extremities of the two chortls, and neglect the terms con- 

 taining the powers of <?-, we shall get, 



y'^ = y' { 1 - -J ( sin - X + sin^ '^' ) j + 2^' «'''^^- 



K 2 From 



