370 THEORY OF HEAT. [CHAP. IX. 



In fact, the function which we thus form, 



gives three terms for the fluxion with respect to t, and these three 

 terms are those which would be found by taking the second fluxion 

 with respect to each of the three variables so, y, z. 



Hence the equation 



v = TT 3 fdn jdpjdq 



y + 



gives a value of v which satisfies the partial differential equation 

 dv _ d*v d*v d*v . 



~dt~dx^d^ 2 + ^&quot; 



373. Suppose now that a formless solid mass (that is to say 

 one which fills infinite space) contains a quantity of heat whose 

 actual distribution is known. Let v =F(x, y, z) be the equation 

 which expresses this initial and arbitrary state, so that the 

 molecule whose co-ordinates are x, y, z has an initial temperature 

 equal to the value of the given function F(x,y,z). We can 

 imagine that the initial heat is contained in a certain part of 

 the mass whose first state is given by means of the equation 

 v F(x y y, z), and that all other points have a nul initial tem 

 perature. 



It is required to ascertain what the system of temperatures 

 will be after a given time. The variable temperature v must 

 consequently be expressed by a function &amp;lt;j&amp;gt; (x, y, z, t) which ought 

 to satisfy the general equation (A) and the condition &amp;lt;/&amp;gt; (x, y, z, 0) 

 = F(x t y, z}. Now the value of this function is given by the 

 integral 



v = 7r 



In fact, this function v satisfies the equation (A), and if in it we 

 make t = 0, we find 



IT 9 fdn j dp (dq e-W^+&F(x, y, z), 

 or, effecting the integrations, F (x, y, z). 



