62 BELL SYSTEM TECHXICAL JOURNAL 



ypT hovers around zero and ^r is nearly equal to — i/'o • Differentiating with 

 respect to t gives 



yp'r = A' cos /3t — .4/3 sin ^r 



yp'^ = (A" - AI3^) cos I3t - 2.4 '/3 sin 0t 



\l/Q = Ao — ^oi8% \po = Ao 



where .4o and .4o are the values of .4 and its second derivative at r equal 

 to zero. These lead to 



B = (AoA" - AAo) cos/3r - 2 AqA' 13 sin 0t 



C = (AA" - A") cos' /3t - AoAo + (.4^ - A)'^' 



We wish to show that C -\- B and C — B are of the same order of magni- 

 tude. If we can do this, it follows that M22 — M23 is much smaller than 

 M22 + M23 since \po — 4^^ is approximately 2i/'o while xj/o + 4'ri is quite small. 

 Consequently we will have shown that M23 is nearly equal to M22 . 



So far we have made no approximations. We now express the slowly 

 varying function A as a power series in r. Since i/'o and \po must be zero 

 for the type of functions we consider, it follows that 



A = Ao + ^-Ao + •• 



A' = tAo + • • 



A^' =A'J + 'jA','^+ .. 



where we neglect all powers higher than the second. Multiplication and 

 squaring gives 



2 j' 2i j" 



A — A'o = T AoAq 



AA" - A'' = AoA',' + ^' Uo.4r - A'o") 



= AoA'o + F 

 2 

 AU" - AA'o = ^ Uo^r - a'o") = F 



Since, for small r, A and A" are nearly equal to ylo and Ao, respectively 

 we see that the difference on the left is small relative to AoAo, i.e., 



\F\ << \AoA'o'\ 



