Yih 



P' = Pi'(y - Ye). 



If pj were not zero p' would be positive for y slightly larger than 

 y . With this realization, it is immediately clear that the bracket 

 in (3 2) has no singularity at y . Let us denote the bracket in (3 2) 

 by the sumbol B, which is a function of y, Q?, and N. Then if m 

 is the minimum of B/p between two points y and y , with 

 < y < y :S 1 , for a = Q , and if 



m > i — I ^ n = a positive integer, (51) 



by tne use of Sturm's first comparison theorem we know that there 

 must be at least n zeros of f between y, and Yz » whatever the 

 value of f(0) and f'(0). (Note that the "p" in m or in (32) is dimen- 

 sionless.) We can always choose f(0) =0. If (51) is satisfied then 

 there must be at least n internal zeros of f. We can increase a 

 so that, again by Sturm's first comparison theorem 



f(l) = 

 for 



where 



tt, < 0^2 < a3< . .. < cc^. 



It is evident that for a = ccj there are at least n- i internal zeros. 



Hence we have 



Theorem 4. Under the assumptions stated in the second paragraph 

 of this section, if (51) is satisfied there are at least n modes with 

 c = Uf; and o- = a, (i = 1 , 2, . , . , n) , and with cx\ increasing with 1. 

 For the i-th mode there are at least n- i internal zeros . 



It is easy to show, by exactly the same approach used by Lin [ 1955, 

 pp. 122-123], which we shall not repeat here, that by varying a^ 

 slightly (now not necessarily by decreasing it, as is in Lin's case), 

 c will become complex. Hence we have 



Theorem 5. Near the neutral modes stated in Theoremi 4, there 

 are contiguous unstable modes . 



232 



