tan 



290 NEW SERIES for the 



To the fides of this equation let the correfponding fides of the 

 equation 



- L W= l l— W-*nlA} 



tan A 2 C tan^-A 2 J 



be added ; then we get 



\-r + - L -*=l( rv-r + — Vt)— |(tan* A + tan 3 lA). 



3 A tan A 8 v tan 3 ~ A tan^ A' 8 V 2 ^ 2 ' 



xt 1 t * 1 + tan A fee A , ~ ., . 



Now -— + r zz — : — — r — zz — -, and fimilarly, 



tan 3 A tanA tan 3 A tan 3 A' J ' 



1 , 1 fec 2 4rA 



k + -A-r - ,^ » al{b tan i A + tan 3 I A = 

 tan 3 7 A tani-A tan 3 ^-A 2 2 



tan - A (1 + tan* - A) = tan - A fee 2 1 A : therefore, by fubfii- 

 2 v 2 ' 2 2 ^ 



tuting, we get 



fee* 7 A fee 2 |A 1 . 1 A r , 1 a 



■ — -z— r = 5 f— r- — - tan - A fee 2 - A. 



tan 3 i A 8 tan 3 £ A 82 2 



24. From this formula, by fubftituting a, -a. x a . . . _^_ 



rf 2 4 2«— ' 



for A, and multiplying the fucceffive refults by the fractions 



11 1 



' 8' 8" ' " ' 8"— v we deduce tne following feries of equations, 



the number of which is n : 



fee 1 a 

 tan^ 





