130 PROCEEDINGS OF THE AMERICAN ACADEMY 



or, substituting, 



dpU 



do 

 Similarly the ratio of the corresponding total differential coefficients is 



du 



^^ = xp (x, y, z). 

 Tp 



Whence, changing the variable and integrating, 



d,n dii 



- = rp(x,y,z) = - 



The partial and total differential coefficients of u taken relatively to v 

 cannot be equal, if u involve any other quantity than v, 



,\ u=f{v) 



In short, when a body is heated in any manner whatever, there 

 must exist a function y(y), the same for all bodies, whose derivative in 

 any direction, when multiplied by a constant depending on the nature 

 of the body, gives the flux of heat in that direction, provided v is found 

 constant along the surfaces g) (a-, y, z) = k, which belong to the solu- 

 tion of Laplace's equation for that particular case. 



The first case open to direct and satisfactory experiment is where a 

 plate of metal is heated at two points, and exposed to the air only at 

 its edges. The isothermals in this case belong to the family A log r^ 

 + B log i\ = k ; 



or r^r.p = c. 



This latter form of the equation shows that the two points need not 

 be heated equally. The solution for three dimensions cannot be readily 

 submitted to experiment ; but the probability that f{v) should satisfy 

 the solution for two dimensions, and fail in that for three, is so slight 

 that it may be neglected. 



Our first experiments were with a small iron plate covered with a 

 mixture of wax, rosin, and paraffine. By this means, one curve was 

 obtained at each heating ; viz., that separating the part of the mixture 

 which had melted from that which remained solid. This method car 



