J^gi INVESTIGATION OF CERTAIN THEOREMS 



To dctcriplne 22. The theorems, which were next propofed to be confi- 

 the figuie of the^jg^gj ^^^ ^^^^ ^^^^ determine the figure of the earth from the 

 earth trommea- ' ^ , , 



fures of the per- meaiures ot degrees ot the curve perpendicular to the meridian, 



pendicular in j^ different latitudes. For this purpofe let D' be a degree of 

 tu4ci. *^"^ ^^ thefe curves, in the latitude ^', and D'^ a degree of one 



of them, in another latitude (p'\ Then c being the compreffion, 

 . as before, we hav^ by § 18. ?«D'==a + c fin ^^', 

 andalfo wD"= a + c fin *?)'V 



Hence m (D'- D'O = c (fin «(?)'- fin *(?''), and 



therefore c zz ^ ^^, - — r-rr.. 



This formula may be rendered more convenient for calcula* 



tion, by confidering that fin *^' = ~ , fo that 



fin *(?)'- fin *?)" = 



1 - cof '^f?'- 1 + cof 20^" 



cof2(?)^^ cof2<?'^ ^ But cof 2(?>^'-. cof 2$'= 2 fin (<f>^ + r) X 

 fin ((f-?", wherefore fin 'ip'-fin ''?" = fin (<!'' + ?") X 



fin I?-- <p"), and c = f.., (^.+"j"'; fi„ '(y- y y 



23. In the fame manner, becaufe mD':=a-]-c fin *?>', by 

 (ubftituting for c, we have 



w(D'-D")fin*^ , 



'"^ =^^+fii.(^'+nxfin(^'-n* ^""^ 

 ?»(D^-Dnfing(?) 



24. Laftly, fince mD^ = o + c fin *$'', 



andwD'''=^ + cfin»<p^ 

 dividing the firft of thefe equations by the fecond, and reject* 

 ing the higher powers of c, we have 



^^= 1 + - (fin *^' - fin 2^'), and therefore, 



c D'' ^ 



«=fin*^-fin^r' Hence alfo 



g"" fin(^ + nxfin(^^-r) ' ''' """'^ conveniently for 

 calculation by logarithms, ^ = p.^^^^^^.";^ ^fi^^^^^^.y 



t?5. We 



