^8 INFESTIGJ'TION of certain THEOREMS 



angle of any one of thefe triangles, to a given angle of any other 

 of them, may be found by trigonometrical calculation. Let the la- 

 titudes be obfcrved at the extremities of this chord, and alfo the 

 difference of longitude ; then, from the nature of an ellipfoid, 

 the length of this fame chord may be exprefled, in terms of the 

 axes a and b, together with the latitudes of the extremities of 

 the chord, and the difference of longitude between them ; and 

 this expreffion being put equal to the length of the chord mea- 

 fured will give an equation, in which all the quantities are 

 known, except a and b. Further, if « = ^ -f <r, and if the faid 

 expreflion be reduced into a feries, with the powers of c afcend- 

 ing, that feries will converge very rapidly, becaufe c is fmall in 

 refpedl of a ; then, for a firfl; approximation, we may reje<5l all 

 the terms that involve the powers of c higher than the firft, by 

 which means we ftiall have a fimple equation of the form 

 ma -f- "'^ = A where m and n are fundlions of the latitudes and 

 difference of longitude, and / is the length of the chord. 



Now, if a fimilar equation be derived from the meafurement 

 of any other chord, thefe two equations will give a and c in the 

 fame manner as in § 6. ; and thus, from the meafurement of 

 any two chords, the figure of the earth will be determined. 



35. The length of the chords, thus meafured, fhould be great, 

 fo that they may, if poffible, fubtend angles of feveral degrees, 

 and their poficion will be moft favourable when one of them is 

 in the plane of the meridian, and the other nearly at right an- 

 gles to it. The numerical computation will be found lefs labo- 

 rious than might be imagined ; but the complete folution of the 

 problem, and the full detail of the inveftigation, I am under the 

 neceffity of delaying to fome future communication. 



There feems to be but one difficulty of any confequence 

 that ftands in the way of this method of determining the figure 

 of the earth. It arifes from this, that the afcertaining the pofi- 



tion 



