io INVESTIGATION of certain THEOREMS 



Wherefore alfo the longer axis of the meridian is to its conju- 

 gate, or a is to b as 300 to 299. 



This proportion agrees well with that which was already- 

 pointed out as the moft probable refult, from the comparifon of 

 fingle degrees, and from obfervations of the pendulum. As 

 thefe conclufions are obtained by different methods, they tend 

 greatly to confirm one another. 



8. From this too it feems highly probable, that the uncertain- 

 ty which yet remains with refpecl to the true figure of the 

 earth will be entirely removed by the meafurement of fome 

 other confiderable arches of the meridian. Such an arch will 

 be furnifhed by the furvey of Great Britain begun by General 

 Roy, and (till continued in a ftyle 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 exaclnefs, it may be neceffary to employ a 

 more accurate approximation than has been done in the prece- 

 ding formula, by retaining the fecond power of c. The equa- 

 tions to be refolved will thus become of the fecond order, but 

 as the unknown quantities can be nearly found by the folution 

 of a fimple equation, the farther approximation to their true 

 values will be accompanied with no difficulty. 



9. Concerning this farther approximation it may be ufeful 

 however to remark, that if C 1 be retained, its coefficient in the 



formula of § 4. will be 7^; (<P + 7- nn 4<P ) J 

 and therefore in the formula of § 5. it will be 

 _L_ fr _ y + El ( f in tf _ f in tf)). 



If then the quantity 



tfe (<P" - <P' + 7 ( fm ^ ~ fm 4 ^ ) ' 



computed for any arch of the meridian, be put = <?, and the 



fame 



