2 INVESTIGATION of certain THEOREMS 



Now z — <p X FH, therefore z zz <p {a — ic + y fin 2 <p) 

 - [a — 2c) J> -f y <p fin 2 <p. But fin *<p = J-"> f ^ t]iere . 



fore z = {a — 2C)$-\--cq> — ^__cof2p, and taking the fluent 

 %~(a — ^)<p — ~ fin 2<p. To this value of z no conflant 



quantity is to be added, becaufe it vanifhes when z~o. 



Therefore an arch of the meridian, extending from the 



equator to any latitude <p, is = a<p — c - y<p -f ~ fin 2<p. ). 



5. This theorem is alfo eafily applied to meafure an arch of 

 the meridian, intercepted between any two parallels of the equa- 

 tor. 



Thus, if MN be any arch of the meridian, <p' the latitude of 

 M, one of its extremities, and <p" that of N, its other extremity, 



we. have AM = a<p' — ~ (<p' -f | fin 2<p' ), and 



AN = af — | (<p" -f- 1 fin 2<p" ) . Therefore the arch 



. MN -a (<p" — <?') —\ ({<?"—<?') + f fin 2<p"— 2fi n 2 <p'). 



6. If, therefore, MN be an arch of feveral degrees of the me- 

 ridian, the length of which is known by actual meafurement, 

 and alfo the latitude of its two extremities M and N, this lall 

 formula gives us an equation, in which a and c are the only 

 unknown quantities. In the fame manner, by the meafurement 

 of another arch of the meridian, an equation will be found, in 

 which a and c are likewife the only unknown quantities. By 

 a comparifon, therefore, of thefe two equations, the values of 

 a and c, that is of the radius of the equator, and its excefs above 

 half the polar axis, may be determined. 



Thus, if / be the length of an arch meafured, m the co-effi- 

 cient of #, and n of c, computed by the laft formula ; and if /' 



be 



