LOADED LINFS AND COMPENSATING NETWORKS 459 



This was vt-rirutl l)y tlu- iik-iIkkI of iiiuIiti'iiiiiiR-d cnotVu iiiits. (on- 

 sisting ill .issuiuing 



x = y + bif-\-b3y' + bty'+ . . . 



.111(1 tluii suhstiiuiiiig this expression for x into (liJ-A) to fx.iki.iti- the 

 b's by treating the resulting ecjuation as an identity. 



In the degenerate case where only even powers of .v are present in 

 (15-A) the formula (1(5-.A) when applied directly does not correctly 

 express the solution (for reasons appearing below). However, the 

 given etiuation, containing only even powers of .v, say 



y=x-+CiX*+C3:i^+CtX^+ (17-A) 



can be correctly solved for (.v-) by direct application of (lO-.A), with 

 a, = c,; and then the value of .v can be expressed as a power series in 

 _v by extracting the square root of the power series representing (jc'). 

 In that way the solution of (17-A) was found to be 



, /715, 143 , , 11 , 11 , 1 \ , , / ^1»« . 



,1105, 195^ 195 ,13 .13 1 \ . , ,,„,, 



-f-gg-cjc, -^(if.-^f2C^ + ^C2fo+-jC3fi- 2<^6Jr+ • • • • (18- •^' 



This result was verified by the method of undetermined coefficients. 

 by writing .v in the form 



x=Vy'{l+eiy+eiy'+e,y'+ . . .) (18.1-A) 



and then evaluating the e's by substituting (18.1-A) into (17-A). 

 Still another method would be to extract the square root of (17-A) 

 as the first step, thereby expressing V y as a power series in x of the 

 form (15-A); and then reverting by application of (16-A), thereby 

 expressing a: as a power series in V y and thence of the form (18.1-A). 

 For use in this connection it may be noted that the square root of a 

 power series having the form 



>" = l+;»,.-c+/j2.r=+/(3.v'+ . . . 



will be of the form 



y=l+k,x + kix- + k^'+ 



The k's can be evaluated by identifying the first equation with the 

 square of the second; their values are found to be . 



*4 = .y;4-i,*i-*i*3. ki = Uii-kik,-kik3, 

 kt = ^ht— hkl — kiki — kiki. 



