VOL. XLVIIl.] 



PHILOSOPHICAL TRANSACTIONS. 



399 



Consequently, the logarithm of 1 -|- n will be expressed by n — -^ n* + ^ n*— 

 •J- n* + -}- n", &c. as above asserted. 



Again, since ~ ^ = 1 -f- n ^ n^ ^ n^ ^ n* -{■ v} -\- &c. as appears by actual 



division. And, since the excess of that series above unity, is the series 

 7J + w'^ + n' + n* &c. 



Therefore the logarithm of —— will consist of the sums of the powers of that 

 series, multiplied by the above-found co-efficients -|-, — ^, -f ^, — -j., + ^, &c. 

 r^l u. ?? r n' + 1n^ + ^n* + 4n\ &c. 

 r? -\- 3n* -I- 6n^ &c. 

 •St \ n* + 4w^ &c. 



L5j*- L n', &c. 



Now the<^U ^f.^ 



And, 



TT 



+^ 



^ 



■* 



n+ n" -f- w^ -I- n* + 



1 



— in — -l-n' — fn — 4-n', 

 + i"' + fn' + ^n\ 



&c. 



&c. 



&c. 



— in' — -J-7r, &c. 



+ i"', &c. 



The sums of which, viz. 71 + in* + -^n^ + ^n" -f -fn', &c. will be the loga- 

 rithm of- , as above affirmed. 



1 — n' 



The operation necessary to find the co-eiRcients of a series, which will express 

 the length of the arc of a circle, by the tangent of that arc, and its powers, is as 

 follows : 



Let a represent the length of the arc, and t its tangent ; then the tangent of 



that arc whose length is 2a, will be^-^; which fraction is equal to the infinite 

 series, 'It -f 2t^ -j- 2t^ + 21^ -{- 2t^, &c. by division. And by performing the 

 necessary multiplications, or divisions, it will also appear, that 

 8t^ -I- 24t' + 48<'' + 80^^ &C. 



M — tt' 



^1 —tt' 

 i--y= 



{---r = 



32t'-{- 160 f + 480/^ &C. 



I28t' -\- 896/', &c. 



512/^ &c. 



Now if we assume, for the value of a, the following series, i + ari' -f- y^ -f. 

 zf + ufi, &c. Then 2t + 2xi^ + 2yt' + 2zf -f 2ut^, &c. = 2a. 



And because — — is the tangent of the arc whose length is 2a, therefore 



J^J^x l-^^f + y i-^^y + z {-^y &c. = 2a. 



