294 N£W SERIES for the 



let the other complex expreflion 4 tan 2 ^ A + 3 tan 4 4- A be de- 

 noted by FyA; and if there were others like it, that is, which 

 only differed by having ^ A, | A, &c. inflead of ± A, they would 

 be denoted by F ^ A, F -i- A, &c. ', then our laft equation will 

 (land thus, 



and fimilarly, putting 1 A, ~ A, i A, &c. fucceflively for A, and 

 multiplying the refults by the feries of fractions —?, —^, —r^y, 



& /♦ A - igs/i A ~ ip + ^3 F i A, 

 &c. 



By continuing this feries of equations to n terms, and then 

 taking their fum, and rejecting what is common to each fide of 

 the refult, exactly as in the inveftigations of the three preced- 

 ing formulae, we fhall get 



+^FtA+ 4aF^A+4,FiA... + -4fJL 

 16 a 16* 4 16? ' 16" 2 n 



and this equation holds true, n being any whole pofitive num- 

 ber whatever. 



2,7* Let 



