14 INVESTIGATION of certain 'THEOREMS 



alfo interfed FR in R, fo that R is the centre, and RF the ra- 

 dius, 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 = <p. Alfo let AC = «, CD = ^, and « — • /< = c, as be- 

 fore. 



, a' cof f 



Then from the nature of the elhpfis, FO = ^^^cof> + A»fmp^ > 

 and becaufe fm FRO : i : : FO : FR, that is, 

 cof?) : 1 : : FO : FR, FR = ;77^^p==Fif^ 5 ^^^ t^'«' ^^^^e- 

 fore, is the radius of curvature of the fedion of the fpheroid 

 perpendicular to the meridian at F. But the radius of curva- 



ture of the meridian at F, that is FH _ ^„'<-ot'<5 + AMinV 

 therefore FR : FH : -^y^^^^T^J^^^^i •• („. cof/+ 5. fi„p.)4' ^"^ 

 dividing both by — — ^, . , ■ > we have 



FR : FH : : rt' cof ip' + 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 : : «' cof (p' + b^ fin <p- : b\ and g- =: p— • 



= fi cof (p^ + fin '<?. Hence 5- — fin <p^ = ^ cof <p' and 2. = 



col cp 



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

 D, D' and q> are known. 



16. To find a and ^ themfelves, if w = 57.2957, &c. oj- 

 the number of degrees in the radius, fo that wD' = FR 



