2«8 NEW SERIES for the 



Put I -=^ = t, and * 7 c °f f ^ = ,, then cof A = j-.', 

 i-fcofA i + cofi-A i -f * 



and l + cofA - _L_ ; alfo cof ± A = *=£-, now cof i A 

 2 I + / 2 I + r 2 



=Vi±i2£A therefore i^r = _^_ 

 2 i -f t ,y /FT? 



and hence 



*' = 



Vi + t — i 

 Vi + t + i 



22. Upon the whole, then, the refult of the inveftigation of 

 the fecond feries may be ffated briefly as follows. Let a de- 

 note any arch of a circle, its radius being unity, then 



i i -{- cof a , i 

 41 — col ~ a 6 



2=1 



J 



_ J 4 i+coft^ 4 3l +cofi;^ 441+cofitf" ' i 

 + T( m ) + T (m + i) + S 



where T<„) and T(«+i) denote any two fucceffive terms of the 

 feries in the parenthefis, and S denotes the fum of all the fol- 

 lowing terms ; and here S will always be between the limits 



, I _, (T( m ) — l6TCm+i)) T(m + t) 



T W ,and- T {m+ i)— (T ( .) — T ( .+,)) ' 



-Li W ,«- I5 



that is, it will be lefs than the former, but greater than the lat- 

 ter quantity. 



The feries of cofines are to be deduced one from another by 

 means of the formula 



„t . ./i-f-cofA 

 cof - A = V— !— 



Or, 



