ALGEBRAIC FORMULA. 281 



fuch, that the difference of thefe arches fhall be equal to a cer- 

 tain affignable ftraight line. 



For, let (2) zz DN, another elliptic arch, reckoned from the 

 extremity of the tranfverfe, and ((p) = KM, the correfponding 

 arch of the tlrcumfcribing circle ; then, if we find (<p'), ((p^Oj 

 ((p'"), &c. a feries of arches fimilar to the feries (p', <p', <p"', &c. 

 (Art. II.) or fuch that 



r / /A J'^ 4(^0 



fin 8(n = — ^^^i^ 



(i + f' ') ^1 _£//' fin '4(;ip«')'' 

 &C. 



and put {&) to denote the hmit of the arches (<p), {<p'), {(p"), &c. 

 alfo, 



it follows, that (z) —£(!)£+ <»(R). 



Now if we fuppofe the arches (p and ((p) to be fuch that 

 <P' + (?>') = \> '^t^en it follows that fin 2<p' =: fin 2((p'), and fin 4(p' 

 =: — fin4(ip'), and fince 



(i + £*) v/i — f'= fi^iT^Ip'' 



alfo, fin (4^0 = — — ^^"4W; " ■ 



> V^f y (i+/'')/i — f''fin'2(?')^' 



we have fin 4<p'''' zr — fin 4(1^'''). Now when tp' =: o, then 

 ((p') =: 2 ; but when (p' — o, it appears (Art. 5.) that cp''' = o, and' 



alfo, that when (?i',),— ^, then (ip*) = ^ ; therefore, when 4<p"' = 0, 



aW) — ^'^- Again, while cp' increafes from o to -, (<p') will de- 



creafe from - to ^; hence, iand from the two laft equation's, it 



follows,. 



