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It may l>o noted that (22-A), when reRarded as a power series in 

 (/>i'), is of the form (l."i-.\) and hence that (/>i-) can he expressed 

 as a power series in X 1)\- diii'ct .ipplicalion of (Ki-A); the result is' 



,,._. X^,4X^ lliX' lOX^ WX^ 



^^' - ^ - 3 + 40 - 943 + 14175 + 93555 ^^^^^ 



In certain applications this fornuda for D\" is more usefid than formula 

 (22) for Du though the two are ultimately equivalent. A formula for 

 p^ is obtainable by dividing both sides of (23-A) by X; for p- = D\- \, 

 by (16). 



An alternative formula for Di can be obtained by starting from 

 (iregor>'s series (20.1-.-\). Application of this to (21-A) enables the 

 left side of that equation to be expressed as a power series in tan D\\ 

 and when the resulting equation is reverted by means of (18-.^) the 

 result is' 



/ X X- IIX^ 1357X^ \ 



tanZ), = v XV'+6" 360 ~5W0+ 1814400 • J- ^^"^-^-^^ 



Series that are even more convergent than (21) and (22), though 

 much less simple, can be obtained by expanding the original function 

 in the neighborhood of a value of the variable known to be an ap- 

 proximate solution of the equation to be solved, and then reverting 

 the resulting series. To formulate the procedure analytically and 

 generally, let « denote the \ariable, and ^(h) the function; and let 

 the equation to be solved for u be 



Hu)=q. (24-A) 



Then, if U is an approximate solution of this efjuation. application of 



Taylor's theorem leads to the following im|)licit c(|uati()n for u— U: 



The left side of this is known. The right side is a power scries in u— U, 

 with U known; the better the approximation represented by U, the 

 more rapidly convergent is the series. This equation (25-A) in u—U 

 is of the form (15-A), with 



and thence (25-A) can be reverted by application of (16-A). so that 

 u— U will be expressed as a power series in iq — 4'iU)] ^'{U). 



