I/O ' Dr. James Bottomley on 



i_ ^ _ _ _ M ■ 

 U^ c^« ~ " '^ U ' 



the integral of this equation is 



^ = "7 ~fimdt ; (51) 



a I i -^ dt +f{x) 



the arbitrary functions ^(/) and f{x) may be so chosen as 

 to satisfy given conditions ; suppose that when ;i;=0,/ = 0, 

 in (51) substitute t^^ for U and then give these values to x 

 and /, we shall obtain 



differentiating and dividing by 



-ff^mdt, 



we find 



- /x^(^) = CI 



by substitution, from (51) we shall obtain 



to determine f {x), suppose that the body A was initially 



uniformly distributed through the cylinder, giving to / the 



value 



P 



and to / the value 0, from (52) we shall obtain 



P 



/(^) = /L^-l; 

 substituting in (52) we obtain as the solution of (50) 



at 

 -/AT) e 



' == 7- (53) 



e«' - 1 + r-"" 



We may now consider a third special case of (15) J 

 in that equation make ^=0, then we get 



ii' = -«,-».* ; (54) 



dxdt fh: 



