266 THEORY OF HEAT. [CHAP. IV. 



the total movement. As the time increases, the partial states of 

 higher orders alter rapidly, but their influence becomes inappre 

 ciable; so that the number of values which ought to be given to 

 the exponent i diminishes continually. After a certain time the 

 system of temperatures is represented sensibly by the terms which 

 are found on giving to i the values 0, + 1 and 2, or only 



and 1, or lastly, by the first of those terms, namely, ~ I da/ (at) ; 



there is therefore a manifest relation between the form of the 

 solution and the progress of the physical phenomenon which has 

 been submitted to analysis. 



282. To arrive at the solution we considered first the simple 

 values of the function v which satisfy the differential equation : 

 we then formed a value which agrees with the initial state, and 

 has consequently all the generality which belongs to the problem. 

 We might follow a different course, and derive the same solution 

 from another expression of the integral ; when once the solution 

 is known, the results are easily transformed. If we suppose the 

 diameter of the mean section of the ring to increase infinitely, the 

 function &amp;lt; (a?, t), as we shall see in the sequel, receives a different 

 form, and coincides with an integral which contains a single 

 arbitrary function under the sign of the definite integral. The 

 latter integral might also be applied to the actual problem; but, 

 if we were limited to this application, we should have but a very 

 imperfect knowledge of the phenomenon; for the values of the 

 temperatures would not be expressed by convergent series, and 

 we could not discriminate between the states which succeed each 

 other as the time increases. The periodic form which the problem 

 supposes must therefore be attributed to the function which re 

 presents the initial state; but on modifying that integral in this 

 manner, we should obtain no other result than 



0&amp;gt; = IT- {&amp;lt;**/ () 2e-** cos i (OL - x). 



ATTJ 



From the last equation we pass easily to the integral in 

 question, as was proved in the memoir which preceded this work. 

 It is not less easy to obtain the equation from the integral itself. 

 These transformations make the agreement of the analytical 

 results more clearly evident ; but they add nothing to the theory, 



