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XXXVII. Reversion of Power Series. 

 By C. E. Van Orstrand *. 



A LARGE number of the series employed in pure and 

 applied mathematics are special cases of the integral 

 power series, 



y = a x + a l x 2 -\-a 2 x :i + (1) 



In the numerous applications of this series, it is oftentimes 

 necessary to express x as a function of y by means of the 

 integral power series 



x = A 0l/ + A 1 if + A 2< i/+ (2) 



The usual method of procedure is to substitute the value 

 of y in the second equation, equate coefficients, and then 



solve for A , A 1? A 2 , ... in terms of a , a 1? a 2 , The first 



three or four coefficients may be determined in this way 

 without much difficulty, but the coefficients of the higher 

 powers of y are so complicated that this method is almost 

 useless for the determination of their values. 



To obviate this difficulty, Professor McMahon f bases the 

 development of the second equation on Lagrange's series. 

 He puts 



z—^- b = — ai - b = — ~ 



Cl CLq CLq 



vV = z +b 1 x 2 + b 2 x 3 + ... —z + (j>(x), 

 n r/1 = n(n + l){n + 2) ...(n + r-1), 



and obtains 







r/l 





hn - 2 + 1 p!g\... 



&?&*..., ... (a) 



as the coefficient of z n ~ x in the reverted equation. The 

 exponents and subscripts of the b's in this expression satisfy 

 the conditions 



p + g + ... =r + l, 



pi + qj+ . .. =w — 2. 



Another method of obtaining the general term of a reverse 

 series has been suggested by Professors Harkness and Morley. 

 They differentiate (2) with respect to x and divide by y n . 



* Read before the Philosophical Society of Washington, D.C., Oct. 9, 

 1909. Communicated by the Author. 



t " On the General Term in the Reversion of Series," Bull. Am. Math. 

 Soc. p. 170, April 1894. 



