10^ INVESTICATIOKOF CEETAIN THEOREMS 



Inveftigabon o' *r •*__,-* • 



formulas for -Now 2 = <p X FH, therefore 2 r= (f) (a - 2c + 3c fin »^) 



computing the l «» r C0(1> 



figure of the = (a - 2c) <p + 3c ^ fin»i. But fin «<? = i — -:, there- 



ear 



ch. 



fore r =r (a — 2c) ^ 4- ~ci~-~cof2<P, and taking the fluent 



(c\ 3c 

 a — - 1 $ fin 2(p. To this value of z no conflant 



quantity is to be added, becaufe it vani ihes when z = o. 

 Therefore an arch of the meridian, extending from the 



equator to any latitude ^, is = a^ f(p -{—^ fin 2^. j. 



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

 the meridian, intercepted between any two parallels of the 

 equator. 



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

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



we have AM = t/^' — ~ { <?)' -j — - fin 2^' }» and 



AN = aip"^ — (<P"'{-^ fin 2<P''\ therefore the arch 



MN = fl ir - <p') -^({r - (p'-) -f ^fin 2f -|-fin2^'). 



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

 meridian, the length of which is known by actual meafure- 

 ment, 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 equation will 

 be found, in which a and c are likewife the only unknown 

 quantities. By a comparifon, therefore, of thefe two equa- 

 tions, the values of a and c, that is, of the radius of the 

 equator, and its excefs above half the polar axis, may be de- 

 termined. 



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

 cient of a, and n of c, computed by the laft formula ; and if V 

 be the length of any otiier arch, m' the coefficient of a, and n' of 

 c, computed in the fame manner, we have ma — nc =:l, 



and m'a — n'c = V. 



__,, n'l — nV mfl — tvV , c m'l — mV 



Whencefl= — ; r;c= — z r'*""- = -77 w- It 



Ttw! — mfn mn — ma a n'i — ni' 



1. raay 



