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 afFea D' — D', or D' — D' only, without paying any 

 re"-ard to the variation of D' in the denominator. There- 

 fore, fmce by § i8. we. have ^ = ^D. ^i -y" *"^ ^S=^^"' ^7 § ^^' 

 c _ P ~~" , if we fuppofe equal errors in detei-mi- 



nine D' A, and D' — D'^ and alfo that thefe are the only 



errors, their efFeA will be the fame, in both cafes, if 2 cof f'^ 

 — fm ('<p' -^ fii'i '^")- Now, if we fuppofe (p" the quantity 

 fought, and add cof <f>'* to both fides of the preceding equation, 

 then 3 cof •<?)' = fm ^<p' + cof ^(p' — fin ^qi'' z= i — fin 'ip' = cof^(p". 

 The latitude (p" therefore muft be fuch, that coi(p/' — Vi X cof<p'. 

 If, therefore, <p' be fuch that cof (p' — ^, the cofine of (p" will be> 

 =: r, and ip* therefore — o. Now, 54°. 44' is the arch of which 

 the cofine =: 4- nearly, therefore, if a degree of the meridianj 



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 ip' tr (cof (p') X /j, we make (p' - 50'. 41', (or 

 any thing lefs than 54°. 44',) ^" will come out impofllble. 



27. It 



