g INVESI'IGATION of certain THEOREMS 



Now 2 = ip X FH, therefore z — (p [a — ic + y fin -<p) 

 — [a — 2c) (p 4- 3^ ?> fin ^<p. But fin -■p — iJH^-if ^ there- 



fore 2 '=■ {(i — 2c)<p-\--ccp — — cofsip, and takmg the fluent 

 z — (n — -) (p — — fin 2(p. To this value of z no conflant 



quantity is to be added, becaufe it vanifl^es when z — o. 



Therefore an arch of the meridian, extending from the 



equator to any latitude (p, is = ^(p — '- \(p + ~ £xm<p.y 



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' — j ((p' -f f fin 2(p' ], and 



AN = af — I ((p" + 1 fin 2(p" ) . Therefore the arch 



.rn^-a {p" — <p') -I {{<?" — <P') + ffm 2<p"- I fm 2<p'). 



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

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

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

 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 eqviation 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-efE- 

 cient of a, and « of f, computed by the lafl formula ; and if I' 



be 



