14 INVESTIGATION of certain THEOREMS 



alfo interfect 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 tk <p. Alfo let AC = a, CD = b, and a — b = c, as be- 

 fore. 



Then from the nature of the ellipfis, FO =: . % * °° „ ,. , . 



* ' </ a coi <p + b % imp" 



and becaufe fin FRO : I : : FO : FR, that is, 

 cof> : I : : FO : FR, FR z£ - /a * cot> + ^ fin<) , > and this, there- 

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

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



■?. wit 



ture of the meridian at F, that is FH zz 



yfa 1 gofVf i*finV 



therefore FR ; FH : rr : r , and 



(a 2 cof p l + V fin <p y (a 1 cof? 1 + 6 l Cnp*) 7 

 i 



dividing both by r, we have 



& J (a* cofp* + b x finp a ) T 



FR : FH : : & cof <p* -f £* fin <p* : £\ 



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 : : a* cof p + fc fin ^ : 5% and g sc « a cofg' + *' fin»« 



2 TV 1 



= ~ cof (p" -j- fin 2 <£>. Hence ^ fin <p 2 = pr cof <p 2 and ~ =- 



coi <£> 



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

 D, D' and <p are known. 



16. To find a and £ themfelves, if m zz 57.2957, &c. or 

 the number of degrees in the radius, fo that mX>' zz FR 



