212 THEOKY OF HEAT. [CH. III. SECT. VII. 



This equation contains, in real terms under a finite form, the 

 integral of the equation -^ + -=-$ 0, applied to the problem of 



the uniform movement of heat in a rectangular solid, exposed at 

 its extremity to the constant action of a single source of heat. 



It is easy to ascertain the relations of this integral to the 

 general integral, which has two arbitrary functions; these func 

 tions are by the very nature of the problem determinate, and 

 nothing arbitrary remains but the function /(a), considered 

 between the limits a = and a = ?r. Equation (a) represents, 

 under a simple form, suitable for numerical applications, the same 

 value of v reduced to a convergent series. 



If we wished to determine the quantity of heat which the solid 

 contains when it has arrived at its permanent state, we should 

 take the integral fdxfdy v from x to x = TT, and from y to 

 y = oo ; the result would be proportianal to the quantity required. 

 In general there is no property of the uniform movement of heat 

 in a rectangular plate, which is not exactly represented by this 

 solution. 



We shall next regard problems of this kind from another point 

 of view, and determine the varied movement of heat in different 

 bodies. 



