ALGEBRAIC FORMULA. 259 



*a*h *: f- L 



J (l-|-£ 1 _2 



ra'B 



(I+e 1 — 2ecof p)^ 



c of ft < 



2E C(>f <p) T 



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

 o to fl> 



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



S - m ^- ««* .. . 



V/i + £* — 2e cofp * 



from this affumption it is evident that when <p zz o, then ^zzo 

 and when <p — *, then \J/ a h~o zz r. 



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

 tion, we get 



j.cof.J,- P cof » 'ipfia'p 



y/l + S 1 — 26C0f^ (i + £*— 2£Cofft)r 



and by reducing the fractional quantities to a common deno- 

 minator, and fubftituting r — cof *p for fin i <p ) the fame equa- 

 tion becomes 



4 cof ^ = ?(' — t co <»( cof g — m 



(l + e 1 — 2e cof <p~)T 



But (i — « cof <p) (cof <p — «), the numerator of the latter fide 

 of the equation, may be otherwife expreffed, thus, 



£{(* — «')" — (1+ « 2 ~ 2scof<p)*} ; hence it follows that 



_ yf(l — £')*--(i+ 8 »-.2ecoi>*V? 

 ^ Cof -4/ <{ " ««(« + £Z — 2£ Cof ^ 





therefore, 



4*(i + e 1 — 2Eco£<py 



2 = 4«*col* , £__ p v A + t ._ a , cof ^ 



and, taking the fluents when <p zz t, 



k&A zz (l J t>) , A> •'+«*— a« cof ^ 



Let- 



