SECT. IV.] ANALYSIS OF THE PHENOMENON. 453 



When \ve examine carefully the process of integration which 

 serves to determine the coefficients, we see that it contains a 

 complete proof, and shews distinctly the nature of the results, 

 so that it is in no way necessary to verify them by other investi 

 gations. 



The most remarkable of the problems which we have hitherto 

 propounded, and the most suitable for shewing the whole of our 

 analysis, is that of the movement of heat in a cylindrical body. 

 In other researches, the determination of the coefficients would 

 require processes of investigation which we do not yet know. But 

 it must be remarked, that, without determining the values of the 

 coefficients, we can always acquire an exact knowledge of the 

 problem, and of the natural course of the phenomenon which is 

 its object; the chief consideration is that of simple movements. 



6th. When the expression sought contains a definite integral, 

 the unknown functions arranged under the symbol of integration 

 are determined, either by the theorems which we have given for 

 the expression of arbitrary functions in definite integrals, or by 

 a more complex process, several examples of which will be found 

 in the Second Part. 



These theorems can be extended to any number of variables. 

 They belong in some respects to an inverse method of definite 

 integration ; since they serve to determine under the symbols 



I and 2 unknown functions which must be such that the result of 



j 



integration is a given function. 



The same principles are applicable to different other problems 

 of geometry, of general physics, or of analysis, whether the equa 

 tions contain finite or infinitely small differences, or whether they 

 contain both. 



The solutions which are obtained by this method are complete, 

 and consist of general integrals. No other integral can be more 

 extensive. The objections which have been made to this subject 

 are devoid of all foundation ; it would be superfluous now to discuss 

 them. 



7th. We have said that each of these solutions gives the equa 

 tion proper to the plisnomenon, since it represents it distinctly 



