68 BULLETIN OP THE 



This calculation has generally been made by means of the 

 series — 



/y»3 /yi5 rp7 



tan-'a; = ar — r + ^ _f + &c. 

 8 5 7 



If we put tan -'x = - or -, we have x=\ or -^ ; the former is 

 ^ 4 6 v/3 



inconvenient because it converges too slowly, and the latter on 



account of the radical expression. 



Many other series may be deduced by resolving the arc into 



the sum of two or more arcs whose tangents are known ; thus — 



!* = tan--l =tan-'i + tan-'l (a) 



= 2 tan-i-*: — tan-ii (&) 



2 t 



= 2 tan-^^ + tan- ^A (c) 



= tan - ' ^ + tan - '1 + tan - ^ ^ (d) 



2 5 8 



— 4 tan-^^— tan-i-J— (e) 



5 239 



= 4 tan~v^_tan-'— +tan-'— - (/) &c. &c. 

 5 70 99 



Any of the numbers become exceeding complicated when 

 raised to high powers ; but these equations have all been used. 

 Clausen used (c) for computing n, and Dase used {d) to 200 

 decimals. Machin used (e) and Rutherford (/). Shanks has 

 also lately used Machin's series for computing n to TOt decimals. 



We have also — 



t' 1 3 



Sin -^x = x^ ^ 4- ^-^^ ^5 4- &c. 



sin^a? , 1.3 . , , . , . 



or, x = ^mx-\- -^-g- + ^I'a ^^" ^ + ^'^' ^^> 



Formula (c) gives 



rt = 2tan-^^ _j-tan-'^ = 2 sin"* ^ +sin">_-^_ {K) 



3 t \/ 10 v/50 



On account of the numbers 10 and 50 in the denominators, this 

 form is more simple than any of the previous ones, but inconve- 

 nient on account of the radical and the complicated coefficients. 



