282 DEVELOP MEN? of a certain 



follows, that in the fame circumftances 4<p* will increafe from 

 o to n, and 4(<p / will decreafe from 2t to it ; and becaufe that 



while <b' increafes from - to -, (<£>') diminifhes from - to o, it 



will alfo follow, that while /$' increafes from n to 2t, 4(<£> // ) 

 will decreafe from ?r to o j and fince during thefe changes of the 

 value of the arches 4<p // , 4((p // ), we have always fin 4<p // zz — fin4(<p'), 



it is evident that 4<p" + 4(<p'0 = 2t, or <p" + ($*) zz |. 



Again, from the two equations, fin 8<p"' = ( I+ ^- )v /°_! < . y , fin , 4 ^ 



fin 8(<p"0 rr (^^y'",^ ,,^, and from the confideration 



that f -f- (/) = ~, we find fin 8<p'" zr — fin 8(p'") ; and hence, 



by reafoning as before, we alfo find £>'" -f- (<£>'") rr ^ ; fo that, up- 

 on the whole, we obtain the two following feries of equations : 

 fin 2<p' zz -f fin 2(<p') <p' -f (<?') ==* f, . 



fin 4/ =s — fin (4PO <p" + (?") = f , 



fin-8f=-fin(8f), *'" + (*>'") = f- 



&c. &c. 



and therefore ultimately & -f if) — -• 



Now we have the elliptical arch DF, or z zz — E -f- eR - y 



alfo the arch DN, or {z)zz^E + e (R). 



Hence DN + DF = z + (z) = li!l^| E+* { R-f (r) ] g 

 but, from the two preceding feries of equations, it is evident 

 that a *'+ (0 * - 1, and <? {r + (R)} ='^£2 f m 2<p '. therefore 

 DN + DF = E + C^±a f m 2<f ; and AN _ DF _£i£t£0 fm ^ 



, e 1 fin 2<p ^finp.cofp 



"" «• 1 —** fin *p A ~"" '/*--** fin V* 



Thus 



