46 MR. A. B. BASSET ON THE MOTION OP 



Performing the differentiation and then integrating by parts, we obtain 



We shall presently show that it is possible to determine F(a), so that F(0) = 0, 

 and F() ""' = when a = oo ; hence the term in square brackets will vanish at both 

 limits, and we obtain 



sin'fl 



. . (8) 



We must now determine the functions x an d F so as to satisfy the surface conditions 

 (2) and (3). 



Equation (2) will be satisfied if 



(9) 

 Equation (8) requires that 



exp - - 



Integrating the last term by parts, the preceding equation becomes 



f V X (a) + F () + aF' (a) + *F" (a) } exp. ( - ) /, . ( 1 U) 



provided, {F (a) + aF' (a) } exp. ( 3 /4i>) vanishes at both limits. This requires 

 that F(0) = F'(0)=0, and that F(a)e- a ' and F(a)e-"' should each vanish when 

 a = oo . When this is the case (10) will be satisfied if 



.,..,,;. (U) 

 Whence by (9) 



7T 



