234 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



This is to be true with the side conditions 



c = fixed (lUb) 



u = fixed (111c) 



u' = (Hid) 



= Xu = X = fixed 



(llle) 



Equation (111) is an identity involving only elementary functions. 

 Thus the relation itself is in a sense elementary. Those who wish to 

 believe it, may consider the theorem proved ; for completeness, however, 

 the proof of (111) will now follow. 



Call the left side of (11 la) X, the right side Y. To determine X, we 

 substitute from (7) and (Hid) 



u = 



m 



with 



M -\- m 



u' + 



M 



M -\- m 



2 /2 



t] = U 



Fig. 15 — Definition of the angles occurring in the proof of the convolution 

 theorem. 



