Yih 



Proof. We have shown that it is necessary only to consider the case 



J(y ) < 1/4. We may consider f only, since the proof for f_ is 



the same, and since the solution is either f or f . Now at y = y^ 



we have f = 0, Near y we have 

 + c 



0^--f^-¥=^[lPanP)"/p'-^|-...]. (48, 



(U - c) z z -* 



Since p' is negative and U' and (pU')' are positive, U" is 

 positive. Thus U - c is greater_than U^ z for z > 0, On the other 

 hand -p'/p is less than (-p'/p)c ^'^^ Y ^ Vc ' si-^-ce (In p) " is posi- 

 tive. We know that for small positive z Q is negative, as can be 

 seen from (48). Hence for any z> the term 



_ Np' 



(U - c)^ 



is less than pJg/z and Q is negative. Equation (48) exhibits the 

 behavior of Q near y , Let the bracket in (47) be denoted by - G, 

 Then sj_nce Q is negative and U - c is positive for y > y^ > and 

 since p' is negative and (pU')' positive, G must be positive for 

 y > y . Multiplying (47) by w and integrating between y and 1 , 

 we have 



(pz^^Ww'), - \ z^>(^w'^ + Gw^) dy = 0, (49) 



where the subscript 1 indicates that the parenthesis is evaluated at 

 y = 1, Note that the integral in (49) is convergent in spite of the 

 simple pole in two terms contained in G -- one of which in Q, as 

 indicated by (48), Equation (49) clearly shows that w(l) cannot be 

 zero. Hence the theorem. 



Another theorem is 



Theorem 2. If P a-nd U are continuous and analytic, with p' 

 negative and U positive, and if U" and (In "p) " are negative 

 throughout, then singular neutral modes are impossible. 



The proof for this theorem is similar to that for Theorem 1. 

 The only modification demanded for clarity is that instead of (44) we 

 should write 



f+(y) = z w^(z) 



with z now defined as y^ - y. The equation corresponding to (47) 

 is now 



230 



