ALGEBRAIC FORMULA. 281 



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

 tain affignable flraight line. 



For, let (z) — DN, another elliptic arch, reckoned from the 

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

 arch of the eircumfcribing circle ; then, if we find (<p'), (p*), 

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

 (Art. 11.) or fuch that 



fin 2(©)= - — _. . Si, \, V V 

 iin 4 (<p ) - (l+ ,/ 0l / I _,- fml2( ^r 



(1 + /") \/l — e»* fin 2 4 (?//)*' 

 &C. 



and put (0) to denote the limit of the arches (<£>), (<£>'), (p"), &c. 

 alfo, 



it follows, that O) -^ E + <?(R). 



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

 <p' -f (<£>') ±s ^ then it follows that fin 2cp' — fin 2(<p'), and fin 4<p' 

 — — fin4(<p'), and fince 



.r ^/^ nn 4© 





(1 + e*) y/x—e'- fnT^'* 



alfo, fin (4^) r: JTj7y7i -*" fin > *W > 

 we have fin 4<p // zr — -fin 4(<p // )« Now when <p' ~ o, then 

 (<p') — J ; but when <p' 5s o, it appears (Art. 5.) that $" zz o, and' 



alfo, that when (<p') — - then (p*) = - ; therefore, when a$' = o, 



4(<p // ) rr 2<r, Again, while <p' increafes from o to -, (<£>') will de- 



creafe from - to - ; hence, and from the two laft equations, it 



follows. 



