HfeLATMfG TO THE FIGi/rE OF THE eARTH. 173 



31. Now, if we would compare the radius of a parallel thus To deftern&Ine 

 found, with a large arch of the meridian, we fliall have by that ^^^u^^^^y c„„. 

 means a determination of the figure of the earth, not lefs to be paring an arc of 

 relied on than that given in the beginning of this paper, ^p /""• ^^ w i 

 The inveftigation is eafy by help of the theorems in § 5 and 6. in fame lat. 

 Let FO be the radius of a parallel to the equator, which palTes 



through F, the latitude of which is (p, and is fuppofed known ; 

 and let FO found by the method juft defcribed be ~r, then^ 



. , a^ co( (p a cof (p* 



as m ^ 4. r zz — - ■ ■^ = -. — ■ =-- ZZ . 



^ v/«* coi'^ +^^ ii" <?* ,X 2c ^ 



^ «-v/l fin (p^ 



a 



according to the method of redudion followed In the pre- 



ceding articles of this paper. Then, becaufe ^i - £f fin (p* 



a 



^ I + - fin ?)2 nearly, we have r zi a cof ?i (1 -|t - fin<2)*)=: 



« cof <?> ^- c fin <f)2 cof <p, or if we divide by cof (p, — ^ zi a 4- 



col <?> 



«fin^». Let-l-=:/, then/ = a4-cfincp)2. 

 col <P ' 



32. Again, if <p' and (p^^ are the latitudes of the extremities 

 of an arch of the meridian, the length of which has been mea- 

 fured, and found 3: /', then, according to § 5 we have 



I'zza ((?'' -^l-l ((^'' - ^') + -^ (fin 2^ - fin 2^0 ) • 



If, therefore, m be the coefficient of a, in the former equation, 



and n the coefficient of c; and if m' be the coefficient of a, in 



the latter equation, and n' of c, we have, as in § 6. 



nU — nl' . mn—ml' 



azz — 7 ;-, andczi — ; --, or fince m ZZ 1, 



7nn — mn mn' — mn 



n'i — nV , m'l — Z' , c m'l - /' 

 or:—; 7-r, andczz -; alfo - ZT -— -. 



33. In this way of determining a and c, the parallel of lati- 

 tude may either interfe6l the arch of the meridian raeafured or 

 not. If it interfedt that arch, this method may have the fame 

 advantage that was taken notice of in another folution, viz. 

 that the whole of the data may be furnithed from the fame 

 fyfiem of trigonometrical operations. Thus, in the furvey of 

 Great Britain, an arch of five or fix degrees of a parallel to 

 the equator might be meafured, and compared with the whole 



length 



