QUADRATURE of the CONIC SECTIONS, &c. £93 



26. The mode of reafoning by which we have found feries 

 exprefling the three fir.ft powers of the reciprocal of an arch, 

 will apply equally to any higher power, but the feries will be- 

 come more and more complex as we proceed, befides requiring in 

 their application the extraction of high roots. In the cafe of 

 the fourth power, however, the feries is fufficiently fimple, and 

 converges fafter than any we have yet inveftigated, while, at 

 the fame time, in its application we have only extractions of 

 the fquare root. On thefe accounts, I fliall here give its invef- 

 tigation. 



I T T T 



Resuming the expreffion :r tan - A : 



* tan A 2 tan 4- A 2 2 



let the fourth power, and .alfo the fquare of each fide of the 

 equation be taken, the refult will be 



+ f — -tan 2 ^A-f-itan 4 i-A 



tan+A "' 16 tan 4 4- A 4tan 2 -i-A 8.4 16 



— _ + * tan' ^ A. 



tan 2 A 4 tan 2 i- A 2 4 



Let the firfl of thefe equations be multiplied by 4, and the 

 fecond by 3, and let the remits be added ; then, reducing the 

 fractions to a common denominator, we get 



Vf-4tan 2 A 1 3 + 4tan 2 i-A 14 , 1 ( -, . . . A > 



Let us, for the fake of brevity, exprefs the complex quanti- 



T 



O I a t- aril A 



ty 4 A — by the fymbol /A, (which is not to be under- 



ftood as the product of two quantities / and A, but as a charac- 

 ter denoting a particular function of the arch A;) and, fimilar- 



l y let 3 + 4tan a -iA be denoted by / ± A, and fo on. Alfo 

 ■" tan 4 A 



/ 

 Oo 2 let 



