8 Mr. Ivory on the Length of a Degree perpendicular 



length of a degree perpendicular to the meridian. According 

 to the Survey, a degree perpendicular to the meridian, at the 

 middle latitude between the two stations, is no less than 61 182 

 fathoms, or about 200 fathoms more than in the spheroid, 

 which has been found to agree so well with all the arcs of the 

 meridian that have been most exactly measured. If therefore 

 we admit that the method of investigation pursued in the Sur- 

 vey is exact, we should have an undeniable proof that the 

 spheroid which represents distances on the meridian so ex- 

 actly, fails entirely in the case of measurements, the extreme 

 points of which are different in longitude. But a little reflec- 

 tion will show that the theorem laid down in the Survey for 

 finding the difference of longitude is not rigorously exact. In 

 the geometrical demonstration of the theorem it is tacitly as- 

 sumed, that a geodetical line drawn between two points in 

 different meridians, is contained in one plane. But 6uch a 

 line has a double curvature ; and the two tangents which ma,rk 

 its initial and final directions are not both contained in any 

 plane passing through the extreme points of the line. There- 

 fore if the difference of longitude, and the latitudes of two 

 points on the surface of a sphere and a spheroid, be the same, 

 it is not strictly true that the sum of the azimuths in one case 

 is equal to the like sum in the other case. The method is 

 only an approximation; and it cannot be confidently relied 

 on until it is proved that it approximates to the truth without 

 sensible error ; which is the more necessary to be done, be- 

 cause the whole investigation turns on very small quantities, 

 a few seconds in the longitude producing a great variation in 

 the length of the perpendicular degree. I have therefore been 

 induced to view the matter in a different light, as in this pro- 

 blem : To find the difference of longitude of two points on 

 the surface of a given spheroid, the latitudes of the points and 

 the length of the chord drawn between them, being known. 

 In solving this problem we may likewise assume that the two 

 points are little different in latitude. 



Let a and a (l—s) represent the axes of the spheroid ; A 

 and x' the latitudes of the two points; y the length of the chord 

 between them ; and, neglecting the square of e, put 



COS X COS k' 



sinx(l-2i) sin*/(l-2t) 



9. = ^l_2*sin*x' " V 1-2* sin 1 a" 



then ap and au will be the perpendiculars drawn from the 

 two points to the polar axis, and a q and a t the perpendicu- 

 lars to the plane of the equator. Put a> for the angle between 



the 



