282 DEVELOP MENT of a certain 



follows, that in the fame circumftances 4?)* will increafe from 

 o to T, and ^{(p") will decreafe from 2^ to w; and becaufe that 



while <p' increafes from - to -, ((p') diminifhes from - to o, it 



will alfo follow, that while /^<p" increafes from if to iv, j\.{<p"y 

 will decreafe from tr to ^ and fince during thefe changes of the 

 value of the arches /i^<p", 4 (<?*), we have always fin 4(p''' :=: — fin4(ip'), 



it is evident that /^(p" + 4((p0 = 23-, or p" + {(p") — \- 



Again, from the two equations, fin 8<p = (^^^,/.-)y/^_^,,.f„^,^^^^ 



fin 8(?)'") == (i+,'>Q/T- !*'iin'4(yO'' ^"^^ ^^°'^ ^^^ confideration 

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



by reafoning as before, we alfo find (p'" + (<?'") — | ; fij that, up- 

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



fin 2(p' z= + fin 2((p') (p' + i,<p') - f. 



fin 4?)*=:— fin (4/) <P" + {.<P") 



fin 8«>"' = - fin (8<p"'), <?'" + (?'") = ^, 



&c. &c. 



and therefore ultimately 6-\-[6)'=. -. 



Now we have the elliptical arch DF, or x = — E + ^R ; 



alfo the arch DN, or (%) = ^^ E + ^ (R). 



Hence DN + DF = z + {z) -li!^^ E+^[R-f (r) | . 

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

 that ^i^i^= I, and ^ {r + (R)} =.'^^^ fin 2<p' ; therefore 

 DN + DF = E + ^^^^ fin 2^ and AN - DF -fH^^n 2(^ 



«* fin 2f __ e'fin^i.cof^ 



"" jy I — f' fm 'p"~ /l — e' fin '9' ' 



Thus 



