122 



MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



2. tan" 1 x = x x 3 + - r> x 7 + . . . . (Gregory's Series) 



x-^ i 



.2n+l 



i 7T I I I I 



4. ta.n" 1 x = H x ^H =-. . 



2 # ^ 3 ^ # 7 



i 7T I I I 1-3 I I-3-5 I 



5. sec" 1 x = ----- -^ -- -r H --- ^-~ -^ - . . . 

 2 o; 2-3 x 3 2-4-5^ 2 -4- 6- 7 x 1 



(2n)l 



2 ^U 2 2 (nlY (2H 

 n=o 



6.42 



32 3-53 3.5-7 4 



2. 



3. (tan- 



TT 



11 



L 2< 1 



+ ^ - 2 2k a + p - a - 2 



a= I \ka=I I 



(Schwatt, Phil. Mag. 31, p. 490, 1916). 



/ 9 'I X 2 <i 2 '4 7 



4. vi - x 2 sm" 1 a; = x 1 x 5 x 7 + . . . 



3 3-5 3-5-7 



00 



3 3-5 3-5-7 



yL'tfn+l 



