22 INVESTIGATION of certain THEOREMS 



Here, agreeably to an obfervation already made, we may, in 

 order to eftimate the error produced in |, in confequence of an 



error in the determination of D', and D% and A, fuppofe the er- 

 ror to affect D' — D 7 , or D' — D /y only, without paying any 

 regard to the variation of D' in the denominator. There- 

 fore, fince by § 1 8. we have£ =■ aD/ ~ ^ , and again, by § 24. 



Z — D"0'7— linv/ if we fu PP ofe equal errors in determi- 

 ning D' — A, and D' — T>\ and alfo that thefe are the only 

 errors, their effect will be the fame, in both cafes, if 2 cof <p'* 



— fin (*<p' — fin 2 <p ff ). Now, if we fuppofe <$/' the quantity 

 fought, and add cof <p' z to both fides of the preceding equation, 

 then 3 cof y = fin 2 <p' + cof 2 <p' — fin Y = 1 — fin *<p' = cof >p*. 

 The latitude qf therefore muft be fuch, that cof <$/' ~ V '3 X cof q>. 

 If, therefore, <p' be fuch that cof <p' — -j-, the cofine of <p" will be 



— r, and (p* therefore 53 o. Now, 54 . 44' is the arch of which 



the cofine == -7- nearly, therefore, if a degree of the meridian* 



and of the perpendicular to it, be meafured in latitude 54 . 44', 

 the comparifon of thefe with one another will give a refult as 

 accurate as if the degree of the perpendicular, in that latitude, 

 were compared with the degree at the equator, and more accu- 

 rate of confequence than if any other degree of the perpendi- 

 cular to the meridian, were to be compared with D'. 



26. Hence, alfo, the comparifon of the degree of the meri- 

 dian, and of the perpendicular to it, in the fouth of England, is. 

 better than if a degree of the perpendicular meafured in that 

 latitude were compared with a degree at the equator. For if, in 

 the equation cof p* =r (cof <p') X V^, we make <p' =z 50 . 41', (or 

 any thing lefs than 54 °. 44',) $" will come out impoffible. 



27. It 



