Mr Wilson, On Convection of Heat 409 



source of heat situated at the origin with v = tu = everywhere, 

 and u constant and positive. In this case the differential equation 



is kV 2 — spu jr- = 0. Assume as a solution of this 6 = Ae~ aX (b (r), 



where r= *</ x 2 + y 2 + z 2 and A is a constant. Substituting this in 

 the equation we get, writing <b for </> (?•), 



r da; A; da; V k j 



Let a = — -£r , when this equation becomes 



2A: 



V 2 </> (r) - a 2 <£ (r) = 0. 



3<fc (r) 

 Substituting now the values of - \ - etc. we get 



d 2 6 2 deb „, . 



a"o + - a - a = °- 



or 2 y or 



The complete solution of this equation is 



,, Ge ar + Be~ ar 



<t> = y~ -. 



where and 5 are constants. 



When r is infinite <b (r) must evidently be zero so that = 0, 

 and the solution of the original differential equation becomes 



„ Ae™ 



Hence, when r is very small, 6 = — and -x- — . 



r dr r 2 



If Q is the strength of the source we must have, when r is 

 very small, 



— 4<irr 2 k -j- = Q, 

 dr 



so that A = j— t , and the solution of the problem is 



47r&r 



for this expression satisfies the differential equation and all the 

 conditions. 



VOL. XII. pt. v. 27 



