142 BELL SYSTEM TECHNICAL JOURNAL 



where 



2 7:1 ^2> 



Z> = (^0 + u){b, ^-v)- g - a% F = Q\2bo - 2g + u + v)/{2D,) 



and Dq denotes the value of D obtained by setting a = Q. When differen- 

 tiating with respect to a it is helpful to note that 



da" ^ ^ \daj -^ ^ ^ da' 



and that only/'(^) = df/dD need be obtained since dD/da vanishes when 

 a = 0. 



In order to reduce the double integral to a single integral we make the 

 change of variables 



. = e^(i„ + u - g)/(2A) - ,,,,^l^t^|/"/ , , 



21(60 + u){bQ + V) — g-\ 



s = Q\bo-\-v- g)/(2Do), F = r+s ^^'^^^ 



d(r, s)/d{u, v) = -rs/Do, 4srDo = Q^Q' - 2g(r + s)] 



The limits of integration for r and ^ are obtained by noting that the points 

 (0, 0), (c«, 0), («>, ^), (0, ^) in the {u, v) plane go into {Q^/{2bG + 2g), 

 W(2bo + 2g)), {Q'/{2bo), 0), (0, 0) (0, (3V(2&o)), respectively, in the (r, s) 

 plane. It may be verified that the region of integration in the (r, s) plane 

 is the interior of the quadrilateral obtained by joining the above points by 

 straight lines. Equation (7.14) may now be written as 



'{s"-gg")(2-2r-2s + Q'/g)-g"Q'/i 



a(x)=// 



QW - 2g{r + s)] 



-r— s 



drds 



2g2 y^ 2g^ ^' 



(7.16) 



where y\ and yi are the dimensionless quantities 



2g-(2-2r^2. + QVg)^^_.^^^^ 

 Q2[Q2 _ 2g{r + s)\ 



2gg~'~' dr ds 

 ►2 - 2g(r + s) 

 It is seen that 



'.=// 

 '-// 



yi = 2gC>-2 \y2+ If e-"--' dr ds\ . (7.17) 



Since the integrands are functions oi r -\- s alone we are led to apply the 

 transformation 



// /(^ + ^) ''•' ''* = [ «/(«) ''^ + £' -^^^ /(») ''^ (7.18) 



