SECT. II.] GENERAL INTEGRAL. 26o 



value of v as a function of x is assigned for a certain value of 

 the time t, it is evident that all the other values of v which 

 correspond to any time whatever are determinate. We may 

 therefore select arbitrarily the function of x, which corresponds 

 to a certain state, and the general function of the two variables 

 x and t then becomes determined. The same is not the case 



with the equation -^ + -7-5 = 0, which was employed in the 



preceding chapter, and which belongs to the constant movement 

 of heat ; its integral contains two arbitrary functions of x and y : 

 but we may reduce this investigation to that of the varied move 

 ment, by regarding the final and permanent state as derived from 

 the states which precede it, and consequently from the initial 

 state, which is given. 



The integral which we have given 



~ (dzf (a) 2e - m cos * (a - a?) 



contains one arbitrary function f(x), and has the same extent as 

 the general integral, which also contains only one arbitrary func 

 tion of x ; or rather, it is this integral itself arranged in a form 

 suitable to the problem. In fact, the equation v 1 =f (x} represent 

 ing the initial state, and v = &amp;lt;f&amp;gt; (x, t) representing the variable 

 state which succeeds it, we see from the very form of the heated 

 solid that the value of v does not change when x i%7r is written 

 instead of x, i being any positive integer. The function 



^ e -i z kt cosl (a #) 



satisfies this condition; it represents also the initial state when 

 we suppose t = 0, since we then have 



(a) X cos i (a x), 



an equation which was proved above, Arts. 235 and 279, and is 

 also easily verified. Lastly, the same function satisfies the differ 

 ential equation -=- = k -5-5 . Whatever be the value of t, the 



temperature v is given by a very convergent series, and the different 

 terms represent all the partial movements which combine to form 



