385 



cases of y= 1/2, 3/2, 5/2, ..., one is able to obt.ain the functions 

 y(x, "V) and h {x,'X) in terms of ele.uentary functions. The solutions 

 for these special values of y_ will be considered in the next sectioHj 

 with the ultimate intention of finding the functions £ and h for V = 

 3/4 by interpolation. 



3 . Solutions with initial velocity distributions of the form (1 + x )~^~ ' 

 It is the purpose of this section to obtain the fun; tions y{x.,Tf) 

 and h(x, H) in terms of elementary functions when the initial velocity 

 distribution is given by 



^'^'^^^^ ^r ' ''= ->-''^^'" (19) 



The entire set of solutions may be obtained from one solution by differ- 

 entiation and integration with respect to a parameter^ introduced into 

 iiq. (19) for 3^ = 3/2 as follows 



The Hankel transform of £q. (20) is 



Q^^Uk,(v) = £l2l , (21) 



? 



The solution of Eq. (3) becoraesi/ 



l/ Bessel functions, by G. N. Watson (2nd ed., Cambridge 1944), p. 384 



68 



