the order of that equation: we prove by the same means, that in the integral of an equation of partial differences the number of arbitrary functions may be less, and change as the integral is developed in series, according to the powers of one or other variable; and when the equation of partial differences is linear, we show that by conveniently choosing this variable all the arbitrary functions may disappear and be replaced by constants, infinite in number, without the integral ceasing to be complete. To elucidate these general considerations, we apply them to examples by means of which we show that the different integrals, in the series of the same equation of partial differences, are transformed into one another, and may be expressed under a finite form by definite integrals, which also contain one or several arbitrary functions. In the single case, in which the integral in series contains only arbitrary constants, every term of the series by itself satisfies the given equation, so that the general integral is found expressed by the sum of an infinite number of particular integrals. Integrals of this form have appeared from the origin of the calculus of partial differences; but in order that their use in different problems should not leave any doubt respecting the generality of the solutions, it would have been necessary to have demonstrated à priori, as I did long since, that these expressions in series, although not containing any arbitrary function, as well as those containing a greater or smaller number of them, are not less on that account the most general solutions of equations of partial differences; or else it would have been necessary to verify in every example that, after having satisfied all the equations of one problem relative to contiguous points infinite in number, the series of this nature might still represent the initial and entirely arbitrary state of this system of material points; a verification which, until now, it has not been possible to give, except in very particular cases. The solution which Fourier was the first to offer of the problem of the distribution of heat in a homogeneous sphere, of which all the points equidistant from the centre have equal temperatures, does not satisfy for example either of these two conditions; it was no doubt on this account that the members of the Committee, whose judgement we mentioned above, thought that his (Fourier's) analysis was not satisfactory in regard to generality; and, in fact, in this solution it is not at all demonstrated that the series which expresses the initial temperature can represent a function, entirely arbitrary, of the distance from the centre.
For the use of these series of particular solutions, it will be necessary to proceed in a manner proper to determine their coefficients according to the initial state of the system. On the occasion of a problem relative to the heat of a sphere composed of two different substances, I have given for this purpose in the Journal de l'Ecole Polytechnique, (cahier 19, p. 377 et seq.,) a direct and general method, of
