372 Mr. C. E. Van Orstrand on 



contains odd powers only, 



& 1 =& 3 =& 5 = ... =0, 

 and the reversed series is 



+ [b s + 10b A + 56/ + 55& 2 2 6 4 + 556 2 4 ]s» 

 + [6 10 + 12&A + 12&A + 78& 2 2 & 6 



+ 78&A 2 + 3646 2 3 6 4 + 273W> U 

 + [&!,+ 146Ao + 146A + 7V 



+ 105& 3 2 6 8 + 210b 2 bJ) Q + 356/ 

 + 5605 2 3 6 6 + 840 W + 2380Z> 2 4 6 4 

 + 14286 2 6 ]~ 13 



Again, if the series proceeds by alternate odd powers- 

 beginning with the first, 



b, = b 2 = h s = b r = h = b 7 = b^ ... =0, 



and the preceding series reduces to 



^=^ + & 4 ^ + (& 8 + 5& 4 2 > 9 + (6 12 + 14&A + 35V)2 13 f ... . 



An important case arises when the number of coefficients 

 which do not vanish is finite. The reversed series is then 

 an expression in terms of an infinite series for one real 

 root of a polynomial of the nth. degree *. The first terms 

 in the solution of the quadratic, cubic, and biquadratic are 

 given below. 



(a) Solution of quadratic. 



y = a Q .v + a^x 1 



z ■=■ x — bi% 2 



x=z + M 2 + 26 1 V + 5 Vxr* + U&iV + 42& 1 5 s 6 + 132^% 7 



+ 429^V + 1430-^V + 48626 1 9 .<r 1 ° + 16796&J 10 * 11 



+ 58786& 1 11 £ 12 + 208012 W 3 + 



* Joseph B. Mott, " On the Solution of Equations," The Analyst, 

 vol. ix. 1882, p. 104. Merriman and Woodward, ' Higher Mathematics, 

 p. 27. 





