146 PROCEEDINGS OP THE AMERICAN ACADEMY 



have been arlderl to the element. Let Q be the total amount of heat 

 in the molecule, then 



If Laplace's Operator is written " — V^," 



-^ = — cV-f(v) dxdydz. 



Let s be the specific heat of the body which Dulong and Petit have 

 shown to be a function of the temperature, and lets = ^\v), then 



d^Q ^ d^v . "^'s . dxdydz, 



^^(^) 4f = - "^'-^(^^ 

 If r"^ = x^ -{- ip' and (jp = tan"~^ — 



— \/Hiv\ = ^!^ -J- ^'^") I ^/Tf) , ^y» 



-^ ^ ^ df^ "> r^ . cl(t>-^ "I" r . dr "^ dz^ 



•' '^'^''^ ir = + ' [d7^ + 77dF+ ?W^ + d7^)-^^'^ 



If the body has reached its final state, the element loses as much heat 

 in any given time as it gains, so that f(^v) must satisfy Laplace's 

 Equation, or 



vy(r) = 0. 



Consider a thin plate of metal of practically infinite extent, and 

 of which all points are at a uniform temperature. Let this plate be 

 laid upon and covered witii some perfectly non-conducting material, so 

 that there can be no flux of heat perpendicular to the [)lane of the 

 plate, and let a single point be heated by means of a copper wire 

 pushed through the non-conducting material upon which the f)late lies. 



There will be a fiux of heat from tlie heated point in all directions 

 in the plane of the plate ; and, if the plate is homogeneous, the flux 

 will be the same in all azimuths. 



After the plate has reached its final state, the amount of heat added 

 to each element of the plate will be the same that flows out of it, and 

 dQ = 0. If the plate lies in the coordinate plane xy, there will be 

 no flux in the direction of the axis of z, and hence 



d'f{") — . 

 -d7^ ^' 



