20 INVESTIGATION of ccrtmn THEOREMS 



quantity, when multiplied into - fec'(p, gives the variation or er- 

 ror in -. which error therefore increafes, cateris paribus, as the 



fqoare of the fecant of the latitude, fo that, on approaching the 

 pole, it increafes without limit, and is ultimately infinite. Com- 

 parifons of this kind may therefore be expelled to give refults 

 the more accurate the nearer they are to the equator, under 

 which circle they will be the mod accurate of all. Here, 

 again, however, another circumftance mud be taken into con- 

 fideration, viz. that the method of afcertaining the differences 

 of longitude by the convergency of the meridians, fo conve- 

 nient in furveys of this kind, is applicable only in high latitudes. 

 In a trigonometrical furvey, therefore, of a country lying much 

 fartlier fouth than Britain, a different method of afcertaining 

 the longitudes of places muft necefTarily be adopted. 



22. The theorems, which were next propofed to be confider- 

 ed, are thofe that determine the figure of the earth from the 

 meafures of degrees of the curve perpendicular to the meridian, 

 in different latitudes. For this purpofe let D' be a degree of one 

 of thefe curves, in the latitude <p\ and D" a degree of one them, 

 in another latitude <p". Then c being the comprefrion, as be- 

 fore, we have by § 1 8. »?D' — a -\- c ^n ^<p\ 

 and alfo wD" = ^ + c fin ^p". 



Hence m (D' — D") = c (fm '(p' — fin "-?>"), and 



therefore c =:. j^»S — ^ ■ , „ . 



nn ^ — lin p 



This formula may be rendered more convenient for calcula- 

 tion, by confidering that fin'<p' — ' ~^° ^"^ , fo that 



fin y— fin y = I-C0f2P'-I+C0f2»" __ C0f2y"-C0f2^'^ g^^ 



cof 2(p^'' — cof 2(f)' = 2 fin {<p' + (p'') X fin (<p' — (p*), wherefore 



