£"'ffl 



396 Prof. L. Natanson on the 



Substituting (2), (3), (8), and (10) in (1), we find 



\_-Pi -^ - ^ + pA+ Mte( w 2-- M i) ]&*i 



+ [.... J^ 1+ [....]^ 



+ L ~ p *~fit " S? + p2X2 + A w ' ^ Wl ~~ W2) J S ^ 2 



+ [.. ..]«# + [....]&*- 

 which shows at once that 



^ c/y cfe -^ 



^ =0, (H) 



Pi ~i^ + b? =^ x i +A ^( t '2- w i)) &c - 



P2 



dw 



2 + £2 _p 2 X 2 + Ap 2 pi{Ui —U 2 ), &C. 



d£ B^2 



(12) 



(13) 



These equations have been established long ago by Maxwell 

 and Stefan. 



§ 12. Conduction of Heat. — Fourier's equation of conduc- 

 tion of heat appears to beloug to the class of conservation of 

 energy equations. At first let us avoid employing " normal " 

 variables. Since the motion of the medium and the inter- 

 vention of extraneous forces are immaterial for conduction of 

 heat, we may put 8T = 0, 2P,^=P, and d°Q = ; therefore 



aU = 8Y, and 



or 



] 1 ^{-au+a , Q}=o ; (i) 



(2) 



Hence, in any real transformation, we have 



dt^dt-' 



that is to say, in " normal " variables : — 



d$ dt ~d qi dt + dt " ' ' 



This is the general form of Fourier's equation ; usually 

 dU/d^is assumed to be of the form doc dy dzpc v in an element 

 do? dy dz, p being the density and c v the well-known thermal 

 capacity; and the remaining 'd\J/'dq i are usually neglected. 

 We shall reconsider the present case from a different stand- 

 point in § 19. 



(3) 



(4) 



