OF ARTS AND SCIENCES. 147 



and, if the plate is homogeneous, 



Therefore /(r) must be a solution of the differential equation 

 or f{v) = J + i? log r, 



and the flux = — c 



clf\v) ^cB 



dr r 



Consider a second plate of metal in every way like the first, only that 

 it is heated at two points by means of a Y shaped piece of copper 

 which is itself heated at its stem. The two arms of the Y are pushed 

 through the non-conducting material and are of equal lengths, so that 

 the two points shall be equally heated. 



If Tj and r^ are the distances of any point from the two heated 

 points, it is evident from the theory of conjugate functions that 



f(v) is constant along any curve of the system [j\r..^ = const.] If a 

 is the distance of the heated points from each other, the equation of 

 the system of curves for any one of which /'(y) is constant may be 

 written 



(x^ + /)((x-ar + /)=!'. 



Before the plate is imbedded in the non-conducting material, let it 

 be covered with a thin layer of a mixture of parafRne, rosin, and wax, 

 and after it has been heated long enough to have sensibly reached its 

 final state let the source of heat be removed; then, if there is a 

 clean line of demarcation between the wax that has been melted and 

 that which has not, the form of one of the isothermals can be studied 

 at leisure. Wherever y(r) is constant, v must be constant unless f(^v) 

 is an equation of an infinitely high degree, which is inadmissible; and 

 conversely, if v is constant along any curve, /(y) must also be constant 

 at all points on that curve. If the isothermal traced by the melted 

 wax is a curve whose equation is i\r^ = c, it will be safe to assume 

 that the flux of heat in the intei'ior of a body is 



_ c "^^^"^ 



