ALGEBRAIC FORMULA 273 



But fo* 2 ? — f m ^ 2®' therefore alfo 



J +*?' 1 + 2f'Cof 20 TJ 



Pcofag J ^ fin * 2 ^ - <£' cof 2<p f . 



.y/j +e <* + 2e'cof 2<p' y'l +/ i +2e'cof2^ 



Let us now reduce the terms on the, left-hand fide of the firft 

 fluxionary equation to a common denominator, and we find 

 ^i + ^cof^)(^+cofy) _ j, cof 2(p , f 



(1 +e' 1 + le 1 cof2p)^ 



But from our aiTumed equation, we find 



e' + cof 2tp r 



— y — — , 7 — » — COl 2(2). 



y i ■+- e + 2<? coi 2<p r 



Hence, and from the laft fluxionary equation, we have 



I -f e' 1 + 2f cof 2<p ~ " ^ * 



Again, taking the fquare of both fides of our afTumed equa- 

 tion, and multiplying by e'\ we have ^ ~ ff^ y = e'* fin *2<p\ 

 and therefore 



1 + e' cof 2<p y r—fi — 7* 



Tz , r ■ — vi — <? 2 fin 2 2<zT. 



Let each fide of the laft fluxionary equation be now divided: 

 by the correfponding fide of the laft equation ; thus we find 



<P v 



y^i +e ri + le'colstp 1/ \ -^e' 1 fin-2p' A * 



and by comparing this refult with the fecond fluxionary equa- 

 tion, we alfo find 



acof2<? . «'ip'fin»20' ', P L -r 



— -, = <p COf 2<p. 



y 1 -f e< z + le'coi 2<p y^i — ^ i fin 1 2?> / 



Let us next affume <?* rr ?r 4 f . 2 , then it follows that v/i — <?* 

 = ?—-*,. and e' zz ~~, r ~~\ ; if we now fubftitute I — i fin 2 <2> 



1 + e ' I + v" 1 — e 



for cof2<p, and <? a for , ^y i "l the exprefllon i-f-/*-f-2£'cof2<z>, 



it is transformed to (i-f<?') z (i — ^fin 2 <Z)), fo that, after due 



reduction 



