ALGEBRAIC FORMULA 273 



But ^ ,.^"''^ ,- , = fin ' 2<?)', therefore alfo 



I -f-e" + 2f' col lip ^ ' 



(p cof 2(p I e'lp fin '2^ 



=^ = (p'cOf 2(p'. 



y'l +^'= +2e'cof 2(p' y'T+V' + lTcon^ 



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

 fluxionary equation to a common denominator, and we find 

 Ki + /cof^O(.'+cof2y) _ ^ c^^2p'. 



(1 +f'* + 4e'cof2p)i 



But from our aflumed equation, we find 



e' + cof itp 



■/ ',,,, , r -- . = col 2®. 

 V 1 4- e + 2i' cof 2(p ~ 



Hence, and from the laft; fluxionary equation, we have 



<p(l+e'coC2<p') __ ', 

 I -f «'• + 2f coflp ^' 



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

 tion, and multiplying by f'% we have /^^/.~ ^^°f ^^f ^^ = e'' fin ^2(p', 

 and therefore 



, , „ , , f ■ — y/i — e'^ fin "2®^. 



V I + e" + 2i? col 2ip ^ 



Let each fide of the laft fluxionary equation be now divided 

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



f 1^ 



and by comparing this refult with the fecond fluxionary equa- 

 tion, we alfo find 



y cof 2^ , ^f'fina2(P' *, -3' , 



v/H-«''+2*'cof2?> '^ '/i — e'^-ii^W' — 'P col 2(p. 



Let us next afllime <?^ zz tj ^ -, then it follows that v/i — <?-^ 

 = 7-;— /, and e' z=l ^~^lHL - - if we now fubftitute i — 2 fin ^ip 



for cof2(p, and e" for /-^xTyj ^^^ theexpreflion i+f'^4-2f'cof2(p, 

 it is transformed to (i -j- e'y (i — e^" fin '?>), fo that, after due 



redudlion 



