a) 
1 fima 2 fin +a 
a2 eon 
300 NEW SERIES for the 
31. THE feries we have juft now found, may be prefented 
under various forms. Thus, by confidering that 
To: .. COA hee SU A eOiAe sn TI 2. A 
tanA” finA 2finA ~ 1r—cofl2A’ 
and that 
fin A 
tan A’) cote ein A rok A 1, . di 
3—tan A tn A ~ 24col-A—1~ 21+ 2¢0i2e 
cof A 
it will appear that by due fubftitution the feries may be other- 
wife exprelffed as follows : 
+5 fin; 4 2 fns5a £1 
3 I-+-2c0f + a I+2c0ka ' 331+2c0f 4 
And other forms might be given to it, but they would all con- 
verge with the fame quicknefs, and each term would be lefs 
than ;th of the term before it. The feries, however, under 
whatever form it be given, and all others which like it require 
for their application the trifection of an arch, are, when com- 
pared with thofe we formerly inveftigated, of little ufe as prac- 
tical rules ; becaufe it is well known that to determine the fine, 
or other fuch function of an arch from a function of its triple, 
is a problem which produces a cubic equation of'a form which 
does not admit of being refolved otherwife: than by trials, or 
by infinite feries, both of which procefles are fufficiently labo- 
rious, and only to be employed where the objec in view can- 
not be attained by eafier means. 
32. As from the different feries we have found for the redti- 
fication of an arch of a circle, the {pirit of our method muft be 
fufficiently obvious, I fhall not inveftigate any others at pre- 
fent. 
