124 THEORY OF HEAT. [CHAP. II. 



whence 



&amp;lt;Pv (Fv__d*v (fdr\* (dr\* .dvfd^r dfr 

 dy* + dz* dr 2 \\cty) + \dz) + dr \dy* + 



In the second member of the equation, the quantities 



dr dr d*r d*r 

 Ty Tz ~dtf 2? J 



must be replaced by their respective values ; for which purpose 

 we derive from the equation y z + z* = r z , 



dr fdr\* d*r 



yT-r and 1=^-1 + r -j , 

 dy \dyj dy* 



dr fdr\* d*r 



z = r-j- and 1 = +r- r - , 

 dz \dzj dz 



and consequently 



The first equation, whose first member is equal to r 2 , gives 



the second gives, when we substitute for 



fdr\* /AY 

 \dy) + (&) 

 its value 1, 



If the values given by equations (b) and (c) be now substi 

 tuted in (a), we have 



(Fv d?v dh Idv 

 dtf + dz*~dr t + r dr 



Hence the equation which expresses the movement of heat 

 in the cylinder, is 



dv_J?i(d^) ldv\ 

 dt ~~ CD Ur 2 * r dr) 

 as was found formerly, Art. 119. 



