SECT. VIII.] EQUATIONS APPLIED TO A SPHERE. 125 



We might also suppose that particles equally distant from 

 the centre have not received a common initial temperature ; 

 in this case we should arrive at a much more general equation. 



156. To determine, by means of equation (A), the movement 

 of heat in a sphere which has been immersed in a liquid, we 

 shall regard v as a function of r and t ; r is a function of x, y, z, 

 given by the equation 



r being the variable radius of an envelope. We have then 



dv dv dr , d z v d z v fdr\ z dv d*r 

 j- -y- -r- and -r- 2 = -i-g ( -=- ) + -y- -= , 

 au; ar dx dx dr \dxj dr dx 



dv dv dr d z v_d z v/dr\ 2 dv d~r 



dv _ dv dr , d 2 v __ d*v /dr\ 2 dv d*r 

 ~ a ~ + 



Making these substitutions in the equation 



dv_Jt_(d*v d*v &amp;lt; 

 dt~ CD(dx* + dy z + 



we shall have 



dv K &amp;lt;Pv (dr\* dr\* dz\* dv (d 



The equation x* + y 2 + z 2 = r 2 gives the following results ; 

 dr dr z 



dr . fdr\* tfr 



y r ~r~ an d i = I -j- ) + T -;-= 



z 



j 



d-y \dy] 



dr fdr\ z tfr 



z r-^~ and 1 = -^ + r -j-$ . 

 dz \dzj dz z 



The three equations of the first order give : 



