456 THEORY OF HEAT. [cHAP. IX. 



arbitrary form, and coincides with the axis of abscissas in all the 

 rest of its course. 



This motion is not opposed to the general principles of analysis; 

 we might even find the first traces of it in the writings of Daniel 

 Bernouilli, of Cauchy, of Lagrapge and Euler. It had always been 

 regarded as manifestly impossible to express in a series of sines 

 of multiple arcs, or at least in a trigonometric convergent series, 

 a function which has no existing values unless the values of the 

 variable are included between certain limits, all the other values 

 of the function being mil. But this point of analysis is fully 

 cleared up, and it remains incontestable that separate functions, 

 or parts of functions, are exactly expressed by trigonometric con 

 vergent series, or by definite integrals. We have insisted on this 

 consequence from the origin of our researches up to the present 

 time, since we are not concerned here with an abstract and isolated 

 problem, but with a primary consideration intimately connected 

 with the most useful and extensive considerations. Nothing has 

 appeared to us more suitable than geometrical constructions to 

 demonstrate the truth of these new results, and to render intelli 

 gible the forms which analysis employs for their expression. 



14th. The principles which have served to establish for us the 

 analytical theory of heat, apply directly to the investigation of the 

 movement of waves in fluids, a part of which has been agitated. 

 They aid also the investigation of the vibrations of elastic laminae, 

 of stretched flexible surfaces, of plane elastic surfaces of very great 

 dimensions, and apply in general to problems which depend upon 

 the theory of elasticity. The property of the solutions which we 

 derive from these principles is to render the numerical applications 

 easy, and to offer distinct and intelligible results, which really 

 determine the object of the problem, without making that know 

 ledge depend upon integrations or eliminations which cannot be 

 effected. We regard as superfluous every transformation of the 

 results of analysis which does not satisfy this primary condition. 



429. 1st. We shall now make some remarks on the differen 

 tial equations of the movement of heat. 



If two molecules of the same body are extremely near, and are 

 at unequal temperatures, that ivhich is the most heated communicates 



