108 INVESTIGATION O-P -CERTAIN THEOREMS 



by mcafurement earth will be entirely removed by the meafurement of fome 

 confiTe'rabt" ^^^"^^ confiderable arches of the meridian. Such an arch will 

 arcs. be furnilhed by the furvey of Great Britain begun by General 



Roy, and ftill continued in a flyle of accuracy fo much fuperior 

 to any other fyftem of geometrical operations that has ever yet 

 been executed. In drawing the conclufions from obfervations 

 made with fuch exacinefs, it may be necelfary to employ a 

 more accurate approximation than has been done in the pre- 

 ceding formula, by retaining the fecond power of c. The 

 equations to be refolved will thus become of the fecond order, 

 but as the unknown quantities can be nearly found by the folu- 

 tion of a fimple equation, the farther approximation to their 

 true values will be accompanied with no difficulty. 

 Farther approxi- 9' Concerning this farther approximation it may be ufeful 



mation to be however to remark, that if c* be retained, its coefficient in the 

 then employed* ^ 



formula of § 4 will be --- | ^ -f T" ^i" ^-^ ) ; 



and therefore in the formula of § 5 it will be 



JL (<p'f ~ ^' 4.il (fin ^f - fin 4^)Y. 

 16a \ 4 / 



If then the quantity 



JL ((^»-<^' J^— (fin W - fin 4r)\ 

 J 6a \ 4 / 



computed for any arch of the meridian, be put z= g, and the 



fame, computed for any other arch, be iz^', the equations of 



^ 6 will become. 



a 



ma — nc 4" ^ — ~ ^ ^"^ 



rrc 



.2 



ma — n c -f- ^^ ZZ I , 



a 



10. Here if we put d for the value of a, as given by the for- 

 mula ■■ ; and /* for the value of c, as given by the for^ 

 mn' — mn 



njula — ; — , alfo tfor the corredlon to be made on d, and u 



mri — m'n 



for the correction to be made on h, (o that azud -\- v, and 

 c iz A -|- u, by fubflituting thefe values of a and c in the two 



laft equations we have mv — nu-\- "-- — |j— - — z= 0, and 





Hence 



