116 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



The equations of motion and of continuity (1.7-1.12) may be satisfied 

 by introducing a single quantity, ^, such that 



V = dx dzV (2.4) 



ii = -J - /3 d, ^2^ (2.5) 



m 



zi = —j — di di\p (2.6) 



m 



yi=-d, d^Dyp (2.7) 



m 



112=- di d^Di^ (2.8) 



m 



Pi 



m 



poiD' - ^') dirl^ (2.9) 



P2 = -- Po(i)' - n di'rl^ (2.10) 



m 



Then, if we introduce the symbol, 12, for co — jSuo 



yi + y^ = 2j-d,d2D^yp (2.11) ' 



m 



h- Z2 = 2j - di diUiD^ (2.12) 



m 



PI - P2 = 2j- po{D' - l3')uiQDi^ (2.13) 



m 



It is clear that if 



Drjy = D^xl^ = y = ±yo (2.14) 



the conditions for elastic reflection will be satisfied. The equation satis- 

 fied by rf/ may now be found from Poisson's equation, (1-13), and is 



{D' - /3^) dx' di^P = '-^{D'- fi'){d,' + di)^l. 



we 



or 



{D' - ^')[{u,'D' + ny + coJiu.'D' - n')] = (2.15) 



which is of the sixth degree in D. So far four boundary conditions have, 

 been imposed. The remaining necessary pair arise from matching the 



