458 BELL SYSTEM TECHNICAL JOURNAL 



desired degree of approximation by first developing the left side of 

 (14-A) into a power series involving D„; and then, by successive 

 approximation or by undetermined coefficients, solving the resulting 

 equation so as to express D„ as a power series in X (that is, "reverting" 



the first scries to obtain the second). 



Digression on the Reversion of Power Series 



Since there will be several occasions here for reverting a power 

 series it seems worth while to digress sufficiently to furnish the requisite 

 general formulas for the reversion of power series:* 



Given y = F{x) developed as a convergent power series in .v, 



y = .v+a2-v=+a3.v'+a4.V+ .... (15-A) 



The coefficient of x has been assumed to be unity because the formula- 

 tion of the reversion is much simplified thereby without any real 

 sacrifice of generality; for, if the coefficient of x were Oi, the equation 

 could be reduced immediately to the form (15-A), either by treating 

 CiX as the independent variable, or by di\iding through In- fli and 

 then treating y fli as the dependent variable. 



The given equation (15-A) expresses y as a power series in x. It is 

 required to revert this relation, that is, to express x as a power series 

 in y. In the present work this was done originally by successive ap- 

 proximation, and was verified later by the method of undetermined 

 coefficients. Evidently the first approximation to the solution of 

 (15-A) is merely xi=y, and thence the second approximation is 

 X2 = y — a2Xi^ = y — a2y''. But the higher approximations cannot be 

 written down thus directly; indeed the labor of obtaining them in- 

 creases rapidly. The work was carried through the sixth a])proxima- 

 tion, with the result: 



x = y+i-ai)y-+i2al-a3)y^+{-5al + 5a2a3-aA)y* 



4-(14a.5-21a;a3+6a204-f3a3-a5)/ 



+ {-i2al + 84ala3-2Salai-28aial + 7a2a!. + 7a3ai-a6)f+ (16-A) 



' Cf., for instance, Bromwich, "Theory of Infinite Series"; Goursat-Hedrick, 

 ".Mathematical .■\nalysis"; Wilson, ".Advanced Calculus"; Chrystal, "Text Book 

 of Algebra." But in none of these references is the reversion carried far enough; 

 moreover, the formulas there obtained do not apply directly to a series containing 

 only even |X)wers — one of the cases in the present application. .'\t considerable 

 labor, by two independent methods, I remedied both of these lacks. Somewhat 

 later I came U|M)n a valuable article by C. E. \'an Orstrand, "The Reversion of 

 Power Series" (Phil. Afag., March, 1910), where the reversion is carried to no less 

 than thirteen terms, but is not directly applicable to series containing only even 

 powers. 



