146 THIRD REPORT — 1833. 



to make it serve to determine the laws of propagation. It is 

 remarkable that this integral is not necessary for solving the 

 problem, although, as M. Poisson has shown in his first me- 

 moir, " On the Distribution of Heat in Solid Bodies," and M. 

 Cauchy in the Notes added to his " Theory of Waves," a solu- 

 tion may be derived from it equivalent to that which they have 

 given without its aid. We may be permitted to doubt whether 

 its meaning is yet fully understood, and to hope that, by over- 

 coming some difficulty in the interpretation of this integral, the 

 problem of waves may receive a simpler solution than has hi- 

 therto been given. Be this as it may, the process of integration 

 adopted by M. Poisson leaves nothing to be wished for in regard 

 to generality. It is easy to obtain an unlimited number of pai-- 

 ticular equations not containing arbitrary functions, which will 

 satisfy the differential equation in question, and to combine 

 them all in an expression for the principal variable (ip), deve- 

 loped in series of real or imaginary exponentials. This will be 

 the most general integral the equation admits of, and (to use 

 the words of M. Poisson,) " there exist theorems, by means of 

 which we may introduce into expressions of this nature, arbi- 

 trary functions, which represent the initial state of the fluid : 

 the difficulty of the question consists then in discussing the re- 

 sulting formulas, and discovering from them all the laws of the 

 phasnomenon. The theory of waves furnishes at present the 

 most complete example of a discussion of this sort." 



In a Report like the present, it is not possible to give any 

 very precise idea of the analysis which has been employed for 

 solving the problem of waves. I have thought it proper to call 

 attention to a process of reasoning which has been very exten- 

 sively employed by the French mathematicians of the present 

 day, and indeed may be considered to be the principal feature 

 of their calculations in the more recent applications of mathe- 

 matics to physical and mechanical questions. To understand 

 fully the nature and power of the method, the works of Fourier, 

 particularly The Analytical Theory of Heat ^ the Notes, before 

 spoken of, to M. Cauchy's " Theory of Waves," and the two 

 memoirs of M. Poisson " On the Distribution of Heat in Solid 

 Bodies," must be studied. I will just refer to some parts of the 

 writings of the last-mentioned geometer, where he has been 

 careful to state in a concise manner the principle of the method 

 in question. There are some remarks on the generality of a 

 main step in the process in the Bulletin de la Societe Philoi7ia- 

 tique*. The note before spoken of in the eighth volume of the 



* An 1817, p. ISO. 



