RELATING TO THE FIGURE OF THE EARTH. Ill 



dicular to the meridian in F meet the lefs axis DE in R. Then T« find the axe» 



R will be the centre of curvature of the circle cutting the me- f ^ fphcroid 



. . irom comparing 



ridian at right angles in F ; for at any point in that circle mde- a deg, of the ^ 



finitely near to F, the diredion of the plumb-line, or of gra- "J^rid. with on* 



vity, as it always pafles through the axis DE, will cut DE in 



R ; it will therefore alfo interfed FR in R, fo that R is the 



centre, and FR the radius, of curvature of the perpendicular 



to the meridian. Let H be the centre of curvature of the 



meridian itfelf at F : draw FO perpendicular to DE, and let 



the latitude of F, or the angle OFR = q>. Alfo let AC = a, 



CD =:bf and a — b =€, as before. 



Then from the nature of the elliplis, FO = 



''*^''^^ and becaufe fin FRO : 1 : : FO : FR, 



v^a» cof *(p -f b"- fin ^* 



that is, cof ?) : 1 : : FO : FR, FR =• 



^a* cof »cp -f- b"- fin <p* 

 and this, therefore, is the radius of curvature of the fection of 

 the fpheroid perpendicular to the meridian at F. But the ra- 

 dius of curvature of the meridian at F, that is FH = 



therefore '" ^n? 



V'a»cof»?'-l-A*fin *(?>' 



a* . c* b^ 

 FR : FH : : i * j. 



(a* cof ^* -1- 6* fin <P*)^ (a* cof?>* + 6* fin 9*)» 



and dividing both by — r- 1. we have 



(a*cof<p*-f 6*fin<P*}^ 



FR : FH :: a* cof ?>* -f b* fin <P* : b^, 



15. If then D be the length of a degree of the meridian at F 

 and D' the length of a degree of the circle at right angles to it, 



D' 



D' ; D : : a* cof ^* -{- 6* fin <p» : b^, and ~ = 



a*cof<?*-f ^*fin<f>* «* f^» . r «.. u «^-. 

 I-TY ^ = IT cof ?>* + fin *^. Hence 



D -^"'^' . 

 cof ^ 

 This laft formula, therefore, gives the ratio of a to i when 

 D, D' and <f are known. 



16. To 



