264 THEORY OF HEAT. [CHAP. IV. 



other function can enjoy the same property. To convince our 

 selves of this we must consider that when the first state of the 



solid is represented by a given equation v 1 =f(x) t the fluxion -y 1 

 is known, since it is equivalent to k ^ \ . Thus denoting by 



# 2 or v 1 -\-Jc -j- 1 dt, the temperature at the commencement of the 



second instant, we can deduce the value of v 2 from the initial 

 state and from the differential equation. We could ascertain in 

 the same manner the values v a , v 4 , ... v n of the temperature at 

 any point whatever of the solid at the beginning of each instant. 

 Now the function &amp;lt; (x, i) satisfies the initial state, since we have 

 &amp;lt;f) (x, 0) =/(#). Further, it satisfies also the differential equation ; 

 consequently if it were differentiated, it would give the same 



values for - , -=f , -=/ , &c., as would result from successive 

 at at at 



applications of the differential equation (a). Hence, if in the 

 function $ (x, t) we give to t successively the values 0, ft), 2o&amp;gt;, 

 3ft), &c., ft) denoting an element of time, we shall find the same 

 values v lt v zi v s , &c, as we could have derived from the initial 



state by continued application of the equation -y- = k -j 2 . Hence 



at doo 



every function ^r (x, f) which satisfies the differential equation and 

 the initial state necessarily coincides with the function &amp;lt;f&amp;gt; (x, t) : 

 for such functions each give the same function of x, when in them 

 we suppose t successively equal to 0, co, 2&&amp;gt;, 3&) ... iw, &c. 



We see by this that there can be only one solution of the 

 problem, and that if we discover in any manner a function ^ (x, t) 

 which satisfies the differential equation and the initial state, we 

 are certain that it is the same as the former function given by 

 equation (E). 



281. The same remark applies to all investigations whose 

 object is the varied movement of heat; it follows evidently from 

 the very form of the general equation. 



For the same reason the integral of the equation -rr = k ^ 

 can contain only one arbitrary function of x. In fact, when a 



