OF ARTS AND SCIENCES. 129 



Our experiments were directed to defermining whether these coudi- 

 tions can be satisfied, and 



F{x,y,z)=f{v). 



When a body heated iu any way reaches a final state, — ■ that is, a 

 state where just the same quantity of heat enters each portion during 

 a given time as leaves it, — the function y(y), if it exists, n:iust satisfy 

 the equation 



</x2 "^ df' "^ dz-i 



One, and only one, solution of this equation corresponds to each set of 

 physical conditions. Since ^', and any function of y, are ct>u!^tant along 

 the same surfaces, if, when the body is in a final state, v is constant 

 along each surface of the family qp {x, y, z) =. k, where cp (x, y, z) is 

 the solution of LajDlace's equation, corresponding to the given physi- 

 cal conditions, then it is always possible to find a function of the tem- 

 perature alone, which shall satisfy Laplace's equation as a function of 

 z, y, z, or, what is the same thing, shall be equal to g) (x, y, z) through- 

 out all space. 



For let tt and v be two functions of x, y, z, such that u = k, and 

 V =. c represent the same family of surfaces, then denoting by dii the 

 total differential of u, and by d^u the partial differential relative to x, 



,*, dxU dyU dzU 



dx rfy dz , . 



d^ ~ d^ ~ d^ — r \^^ y^ -)• 



dr dy dz 



If p is any variable, 



dpU dx'i dpX . dyU dpy dzU dpZ 



dp dx dp dij dp dz dp 



And 



dfv dxV dpX dyV dpy dzV dpZ 



dp dx dp dif dp dz dp 



VOL. XIII. (n. s. v.) 



