332 BELL SYSTEM TECHNICAL JOURNAL 



squared. However, we still have some undetermined coefficients the ti , 

 t2 etc. terms. These we can evaluate by noting that we wish these values at 

 y = a, where a is a fixed distance in the actual device. At this distance 

 the t coefiicients in the expression for y must have such values that the value 

 of y does not change with the value of rj. This can only be true if the 

 individual expressions multiplying each power of j? are each equal to zero. 

 Equating these expressions to zero one can evaluate all of the ^'s. For exam- 

 ple the first term yields 



yo(io)h + yiito) = 



or 



>'i(^o) 



h = - 



yaik 



Introducing these values, differentiating and squaring, one finally gets an 

 expression for (y-)y = as a power series in y, the coefiicients all being of a 

 form easily evaluated for any specified field distribution. Since we have by 

 definition called these coefl&cients Ko , Ki, etc. these values are then 



-^0 = yl 



Ki = 2(yoy - yoyi) 



? 



Ki = {yl — 2yiyi + 2yoy2) - 2yoy2 + -^ 



yo 



This then constitutes the formal solution of the problem. We must 

 now particularize our problem to some specific field distribution and evaluate 

 the y coefiicients. Suppose, for example, that there is a uniform d.c. field 

 (E of equation 1) and an alternating field which varies as some cosine func- 

 tion of distance. Then the latter is 



f{y) = cos 



(t + ^) 



and 



y = - £ -f 7? cos (o)/ -f ^) cos ( — -1- c j 



we must eliminate the y which appears in this expression and replace y by 

 its equivalent 



y = yo + vyi + v^y2 + • • • 



and expanding 



