398 FHILOSOPHICAL TRANSACTIONS. [anNO 1753. 



be equal between themselves ; and consequently the values of the unknown co- 

 efficients may be obtained by simple equations. 



Lastly, since the series, which gives the length of the tangent of the double 

 arc, consists only of the odd powers of the tangent of the single arc, therefore 

 none of its even powers can range with it : now these will not occur in the odd 

 powers of that series ; and therefore the series assumed to express the length of 

 the single arc, whose double is to be compared with the sum of the former, must 

 consist only of the odd powers of that tangent ; and then the series first men- 

 tioned results from the operation, as will appear by examining the same, as 

 hereto annexed. - 



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

 the logarithm of a given number, is as below. 



If the given number be represented by 1 -f- «, thep the following series may 

 be assumed to represent its logarithm : 



viz. n -\- xn^ -\- yn^ -{• zn* -f- wn' ■\- &c. 

 and In -f Ixri^ ■\- lyn^ + Izn* -\- lurv' -\- &c. will represent the logarithm of 

 the square of that number ; viz. of 1 -j- 2n -|- nn. 



But, because In -\- nn is the excess of 1 -j- 2ra -j- wn above unity, therefore its 

 logarithm will be also expressed by 



(2n -\-nn) -\- X {2n -|- nnf -\- y {in -\- nnf -|- z (2n -j- nn)* &c. 

 Now (2n -\- nny = Ann -f An^ -\- n* 



(2w -I- nnf = 8w' -f- 12n* -f- 6n' + &c. 



(2n + nny = l6n* + 32ra' -|- &c. 



(2n + nny = ' 32w' + &c. 



Therefore, 



2n -\- nn = 2n -\- nn 

 X {in -]- nny = Axnn -f- Axn^ -\- xn* 



y{ln + nny= 6yn^ + Xlyn* + Qyn"" -\- he. 



2 {In -{■ nny = l6zn* + 32zn' -j- &c. 



u (2n -1- nny = 32un' -\- &c. 



And the sum of these is equal to the logarithm of the square of 1 -|- n. 

 If an equation be formed, of the co-efficients of n% in each of these expres- 

 sions of the logarithm of that square, then 2x = 1 -|- 4ar ; hence — ±. = x. 



And, by proceeding in the same manner with the co-efficients of n^, n*, n*, &c. 

 and supplying the places of x, y, z, &c. as they arise, by the numbers so found, 

 we shall have 



2y= — i + 6yi hence -\-^ = y; 



2z = — 4- 4- V' + l6z; hence — ± = z; 



2u= i — V + 32m; hence -\- -l. = u; 



