204 NEW SERIES for the 
let the other complex expreffion 4tan’ + A + 3 tan*+/A be de- 
noted by F + A; and if there were others like it, that is, which 
only differed by having + A, ; A, &c. inftead of 4 A, they would 
be denoted by F1 A, F4A, &c.; then our laft equation will 
ftand thus, 
ee ifs. pe ed oF ote Loe 
See ay ee aC iia weet 
and fimilarly, putting +A, + A, 4A, &c. fucceflively for A, and 
ees ; 3 I -U@lIegy 
multiplying the refults by the feries of fractions 76’ 16” 16s 
&c. 
16 16° 16° | 16° 
i + ep 41. Fs ee 2s af, 
geieA reid A 16° Fea, 
&e. 
By continuing this feries of equations to m 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 formule, we fhall get ; 
fA= ase ~ 4 betietie “+t 
gia 
b ange 2 
- +QFIA+ BFPAtS FIA. bak S 
and this equation holds true, iad any whole pofitive num- 
ber whatever. 
276 Lar. 
