Dr. Tiarks on the Longitudes of the Trigonometrical Survey. 365 



the sum of the three angles of a spherical triangle whose 

 angular points have the same relative situation to a particu- 

 lar diameter, which is considered as the polar axis ; that is to 

 say, the same latitudes respectively, and the same differences 

 in longitude. The particular case of this general proposition 

 which is employed by Mr. Dalby, is the one in which two of 

 the geodetical lines are meridians, and where consequently 

 one of the angular points is the pole of the spheroid itself: 

 but it will be easily seen, that from the demonstration of the 

 particular case the truth of the general proposition may be 

 immediately derived. The method used in the Trigonometri- 

 cal Survey requires, that if two points on a spheroid having 

 respectively a certain latitude on meridians forming a certain 

 angle are connected by a geodetical line, the sum of the an- 

 gles of this line with the meridians of the points should be the 

 same, whatever the ellipticity of the meridians may be ; and, 

 accordingly, that it should be equal to the two angles of the 

 spherical triangle the sides of which are the co-latitudes of the 

 points, and the inclosed angle the inclination of the two meri- 

 dians. The inclination of the meridians of the spheroid or 

 their difference of longitude is then derived from the two sides 

 and the sum of the angles opposite to them: viz. the co-latitudes, 

 and the sum of the azimuths. Mr. Dalby's method was first 

 published by General Roy in the Phil. Trans, for 1790, in 

 his own words and with his own demonstration. It would 

 appear that this demonstration has not given general satisfac- 

 tion ; for I have observed that the want of success in the ap- 

 plication of the method which is, indeed, acknowledged on all 

 hands, has sometimes at least partially been ascribed to its in- 

 correctness ; whereas the principle on which it is founded is 

 not only perfectly correct, but neither limited by the length 

 nor the position of the geodetical line to which it is applied. 

 Before I had seen Mr. Dalby's demonstration, I had convinced 

 myself of the correctness of the method, with which I became 

 acquainted through that part of the Trigonometrical Survey 

 published in the Phil. Trans, for 1795, by a demonstration 

 which, although perhaps substantially the same as Mr. Dalby's, 

 yet differs in some respects from it. I hope that this demon- 

 stration will not be considered as perfectly useless at the pre- 

 sent moment, and I shall add to it a few remarks on the cause 

 of the failure of the practical application which has hitherto 

 been made of this method. 



Let 1. be the great semiaxis of the oblate spheroid or the ra- 

 dius of the equator. 

 e the excentricity of the elliptical meridians. 



co the 



