ao INVESTIGATION of certain THEOREMS 



quantity, when multiplied into - fec 2 <p, gives the variation or er- 

 ror in ~y which error therefore increafes, c ceteris paribus , as the 



fquare 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 expected to give refults 

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

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

 again, however, another circumftance muft 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 

 farther fouth than Britain, a different method of afcertaining 

 the longitudes of places muft neceffarily 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 compreflion, as be- 

 fore, we have by § 18. wD' =. a -f c fin 2 <p', 

 and alfo «D" = a -f c fin 4 <p" . 



Hence m (D' — D") = c (fin y — fin y), and 



therefore c - g ffil^,. 



This formula may be rendered more convenient for calcula- 

 tion, by confidering that finy = ' ~"^° , fo that 

 fin y— finy i= I - cof2 ^- 1 +cof2»" __ cofi^'-cofi^ Bu( . 



2- 2 



cof 2<p — cof 2<p' s 2 fin (<p' + <t>') X fin (<p' — <p ff ) y wherefore 



fin 



