ON THE SOLUTION OF EQUATIONS IN 

 FINITE DIFFERENCES*. 



THE partial differential equations which occur in various 

 branches of mathematical physics are, for the most part, of such 

 forms that solutions of them may be obtained without much diffi- 

 culty. As is well known, the great difficulty in almost all such 

 cases consists in the necessity of determining which of all possi- 

 ble solutions satisfies the particular conditions of the problem on 

 which we are engaged. It seems that before the time of Fourier's 

 researches on heat, the course which mathematicians had uni- 

 formly followed was, first to obtain the general solution of the 

 equation of the problem, and then to determine by particular 

 considerations the arbitrary functions which it involved. This 

 course undoubtedly would be the most direct and analytical, 

 were there any general method for determining the form of the 

 functions in question : as, however, there is none, the analytical 

 generality of the first part of the process is in many cases sterile 

 and useless. 



Fourier's methods, which depend essentially on the linearity 

 of the partial differential equations which occur in the theory 

 of heat, consist in assuming some simple solution of the equation 

 of the problem, in deducing from hence a more general solution 

 of it, and in determining successively and by means of particular 

 considerations the arbitrary quantities thus introduced in such 

 a manner as to satisfy all the conditions of the question. The 

 general solution with arbitrary functions does not make its 

 appearance in his process ; and the reason why it is so much 

 more manageable than the other appears to be, that it is far 

 easier to determine arbitrary constants in accordance with certain 



* Cambridge Mathematical Journal, No. XXII. Vol. iv. p. 182, November, 

 1844- 



