i6 INVESTIGATION of certain THEOREMS 



i — rj, fin 2 <p z: o. Here therefore # is indefinite, and may be of 



any magnitude whatever ; and it is evident that this is the re- 

 iult which the formula ought to give : becaufe at the pole, or 

 when <p — 90% the perpendicular to the meridian is itfelf a me- 

 ridian, and therefore the meafurement of the two degrees, D and 

 D', is but the fame with the meafurement of one degree. 



When <p — o, that is at the equator, the circle perpendicular 

 to the meridian is the equator itfelf, and we have then a zz i?iD\ 

 a being determined in this cafe by the degree of the equator 



alone. Here alfo we have j- r= V^y which is known to be 

 true. 



18. The preceding formulas may be rendered more fimple, 

 if we aim only at an approximation, which indeed is all that is 

 neceffary in this inquiry. Since c denotes the compreflion, or 

 fince a — c — b, and therefore a 1 — lac — b z nearly, confe- 

 quently the radius of curvature of the meridian at F, that is 



t^ a z (a 1 — 2ac) a % (a — 2c) 



mD zz r = r c 1 = 



(V — 2acfin<p 1 ) 1 a* (1 fin? 1 ) 1 



(a — ic) (1 — ^^ n V\ or m ® = a — 2C "f- 3 C fm<p 2 . In the 

 fame manner triD' rr a -f- c fin <p\ From thefe equations we ob- 

 tain, rejecting always the higher powers of c, 



mCD' — B') -r^, *»(D' — D) fin cd 1 , c D — D 



c = — - — r *i ■> a — m *-> ^— : — r£ ; and - zz -^-, — r-r. 



2 cof (p 1 ' 2 coi <p ' a 2D cof? 1 



These formulas may be transformed into others a little more 

 convenient for computation, by putting fee ^ inftead of f , , 

 and tan <p 7 inftead of ^p ; we have then, 



c = = (D' - D) fee <p% 



a =s mDl — ^ (D' — D) tan <p\ and 



19. We 



