Il6 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



Applying the transformation to the series on the right, with: 



2m + i x (2m + i) 2 



#o = o, A m = , Am = 7 x-/ ; T, co = 4, a = o. 



m - i (m i) (m + 2) 



CO 



5 = , )m 



T ' 



24 ' 2^ v m(m + 2) (2m + i) 2 ( 2 f + 3)' 



=! 



6.26 Reversion of series. The power series: 



z = x bix 2 b 2 x 3 6 3 # 4 

 may be reversed, yielding: 



X = Z + l2 2 + Ctf? + C&* 



where : 

 ci = bi, 

 c z = b z + 2bi 2 , 



2 ! 3 + 2 + 4! 2 



bf) + 12(361^4 + 6616263 + 6 2 3 ) 



+ 60(26^63 + 3^2 2 ) + 33o6i 4 6 2 + 13 2^, 

 ci = 6 7 + 9(6i6 6 + 6 2 6 5 + 6364) + 45(^i 2 ^5 + 6i6 3 2 + 6 2 2 6 3 + 26^264) 



+ i65(6! 3 6 4 + 6!6 2 3 + 3^263) + 495<V&3 + 26! 3 6 2 2 ) 



+ 1 2876/62 + 42961.7 



Van Or strand (Phil. Mag. 19, 366, 1910) gives the coefficients of the reversed 

 series up to c\^. 



6.30 Binomial series. 





