EXPANSION PROBLEMS. 409 



sion in the first section, and partly by the argument just completed. 

 It is found that if .Tq is any fixed value in the interval < 0*0 < tt, 

 the absolute value of un{xo) is equal to f"^cot(x/.) multiplied by a 

 quantity which may be small for particular values of n, but certainly 

 does not approach zero as a limit when n becomes infinite. It is 

 possible to assign a positive constant Cu, such that 



for infinitely many values of 11. It follows that anMn(.ro) does not 

 remain finite, and 



The statement of Theorem II is valid for the series discussed in the 

 present section. 



III. Differential Equation of Order v with Fewer than v — I 

 Specialized Boundary Conditions. 



If two of the boundary conditions, instead of a single one, involve 

 the point x, the resulting series still show essentially the same char- 

 acteristics, provided that the order of the differential equation is 

 greater than 4, so that the specialized conditions remain in the major- 

 ity. We retain the differential equation (15) of the preceding section, 

 assuming now that j' ^ 5, and associate with it the following boundary 

 conditions : 



kg—i 

 Ws (w) = u^'^s) (0) + £ asj u^'^ (0) = 0, s=l,2,...,v-2, 



(39) ' ^ ° 



/Cj— 1 v—1 



Ws {u) = vS'^s) (tt) + X ^,jU^^) (tt) + Z ^sjU^^^ (0) = 0, 

 7=0 7=0 



S = V — 1,V, 



where 



J' - 1 ^ />:i > /.•2 > • • • > k.-2 ^0, V -1^ K_x > k, ^ 0. 



The differential equation being the same, the asymptotic expressions 

 (17), (18), are still available,^^ and the regions Si and Ti may be de- 

 fined as before. In determining the distribution of the characteristic 

 values, it is sufficient to consider those in the sectors »So and S^v-i; 

 all but a finite number of these will be in the regions To and T^v-i- If 



33 For the present we may interpret the bracket symbol as it was interpreted 

 at the beginning of the preceding section, without the refinement which was 

 introduced later. 



