88 THEORY OF HEAT. [CHAP. II. 



7/7 TTr) j~Z$. ~~ ~nf)&amp;lt;3 vv 



CiU \J U UiOC L/X/AJ 



We shall explain in the sequel the use which may be made of 

 this equation to determine the complete solution, and what the 

 difficulty of the problem consists in; we limit ourselves here to 

 a remark concerning the permanent state of the armlet. 



106. Suppose that, the plane of the ring being horizontal, 

 sources of heat, each of which exerts a constant action, are placed 

 below different points m, n, p, q etc. ; heat will be propagated in 

 the solid, and that which is dissipated through the surface being 

 incessantly replaced by that which emanates from the sources, the 

 temperature of every section of the solid will approach more and 

 more to a stationary value which varies from one section to 

 another. In order to express by means of equation (b) the law of 

 the latter temperatures, which would exist of themselves if they 

 were once established, we must suppose that the quantity v does 



not vary with respect to t } which annuls the term -j-. We thus 

 have the equation 



Ul V fill -I -mif X\f T7-Q TIT &quot;J^V IfSf 



-T~* = ~T7 v &amp;gt; whence v = Me KS + Ne , 



ax AD 



M and N being two constants 1 . 



1 This equation is the same as the equation for the steady temperature of a 

 finite bar heated at one end (Art. 76), except that I here denotes the perimeter of 

 a section whose area is 8. In the case of the finite bar we can determine two 

 relations between the constants M and N : for, if V be the temperature at the 

 source, where # = 0, VM + N , and if at the end of the bar remote from the source, 

 where x = L suppose, we make a section at a distance dx from that end, the flow 



through this section is, in unit of time, - KS , and this is equal to the waste 



of heat through the periphery and free end of the slice, hv(ldx + S) namely; 

 hence ultimately, dx vanishing, 



=L ^ * 



^ &amp;lt;*!. 



IT, irr\ rfjJf 1 



Cf. Verdet, Conferences de Physique, p. 37. [A. F.] 



