ALGEBRAIC FORMULA. 259 



^rt3A = C : 



(l+t' 2EC0f ?)t 



B/ Sip CO 



»cofj> 

 (l +£* — aecofp)"^ 



the fluents to be confidered as generated, while <p increafes from 

 o to 5r» 



8. Let us afllime -^ fuch an arch, that 



Sin ■J/ zz ,- ^ ■ • 



from this affumption it is evident that when <p = o, then 4/ =: o, 

 and when <p = *, then -^ alfo = or. 



Taking now the fluxion of each fide of our aflTumed equa- 

 tion, we get 



-ll/ COf -J/ — ^ ^ ; 



^ ^ ^/l + s' — 2£ COf p (l +£*— 2£C0f^)|- 



and by reducing the fradional quantities to a common deno- 

 minator, and fubfl:ituting r — cof '<p for fln '^, the fame equa- 

 tion becomes 



i cof 4. =z g(ir^L ^°f^)(c°f^-0^ 



(l +e' — 2£ cof ^)t 



But (i — £Cof(p) (cof ip — s), the numerator of the latter fide 

 of the equation, may be otherwife exprefl!ed, thus, 



~{ii — «')' — (i +s'— 2ico{ <pY] ; hence it follows that 

 •4/ cof -^ < ~ '''(' + '\~ *' *^°^^>^ 



therefore, ^ r = ^^hsL^ 1 1 ^•'i + e' -2Ecofp^ 



(i+e'— 2EC0f# (l-£-)'^(l-E')- »■ 



and, taking the fluents when <p =: 5r, 



Let- 



