WILSON. — RECTILINEAR OSCILLATOR THEORY. 123 



of the denominator is legimate in the neighborhood of u = i — |k. 

 The denominator therefore becomes 



9,-, K ( • , 1 \ / , . 1 * ni 



u V 2/V 2u u 



The residue is 



1 



, . 1 Kl 



U + I — - K 



2 M 



— =-(l-; K ). 



t 2i(l + «) 2£ 



u = i — — x 



The second integral in (14) is therefore w (1 — w), which is 27ri times 

 the residue. 

 To calculate the first integral in (14) we may rationalize: 



« u u I 



J—4:K , U 3 U 



4:K 



du. 



w 4 - - - 



K K 



The first two terms of the denominator and the first of the numerator 

 are of the order k 2 , or higher between — 4k and and may be neglected. 

 The result is 



- - \u -f V w ( u + 4k) + 4k log ( V u + 4k + 5u) 



o 



— 4/< 



— 2k — 2/dog (± i), 



where the indeterminate sign and the indeterminate value of the 

 logarithm must be properly chosen. This need not trouble us as the 

 term is imaginary, and its only function is to cancel the imaginary 

 part of tt(1 - i K ). The total value of T + T is thus seen to be r - 2k 

 as previously determined. 



We have next to find the value of x = X at the point d 'arret. This 

 may be written as 



-j; 



u du 



V 2k2k 1 +~ 



