1822.] Mr. Adamson Logarithmic and Circular Series. 261 
ARTICLE VI. 
On Logarithmic and Circular Series. By Mr. James Adams 
(To the Editor of the Annals of Philosophy.) 
SIR, Stonehouse, near Plymouth, Aug. 27, 1822. 
Tue insertion of the following series, &c. in the Annals of 
Philosophy, when you have a convenient opportunity, will oblige 
Your humble servant, 
James ADAMS. 
ee 
From the nature of logarithms, we have 
log. (1 tuysu-e4 ee 4S — &e. and 
2 3 4 
log. (1 ti) sts t+ sp -etin - & By sub- 
ee ae tt tal! Ca ah w= 
( — +) -—5(w=5) +5 (w -3) a) 860s 5 FH icke IDLO a) 
By differentiating equation (1), we have 
I Bid 1 1 
l= (1 + =).— (w+ 5) + (w’ “+ =4 _ (ut + —,) + &e. 
Then by De Moivre’s theorem: for multiple arcs, we have 
1 = cos. u — cos..2 u + cos. 3-u — cos. 4u+ cos. u— 
ROU. sie dle ott PD eree PN Sa rdlalle Sho otha Whe te ws bldte we Sold wiles ste (2) 
‘By addition 7 (1 + u) + 1 (1 fs =) = 1c tai (1 4 uy 
p CAM aU (ut > + 2) = log. 2[ E(u + 5) + 1} = 
1 1 1 1 1 
(w 1h =) ae Be ha) +5 (e+ 5) — &c. 
L(l + cos.u) +12 
ee 
+icos.3u—1cos.4u+ &c. By differentiating this last 
4 
From whence we have = cos. u > +cos.2u 
equation, we have a = tan. 1 wu = 2 (sin. wu — sin. 2'u 
+ sin. 3 uw — sin. 4+ KC.) ., cece cece cece rece ceeecees (3) 
By differentiating equation (3), we get sec.? Lu = 4 (cos. u 
— 2cos.2u +3 cos.d3u —4 cog, 4u + Xe.) 2.2... eee (4) 
Perhaps the following simple method relative to multiple arcs 
may not be unacceptable to the young analyst. By division 
