128 M. POISSON ON THE MATHEMATICAL THEORY OF HEAT. 
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 conve- 
niently choosing this variable all the arbitrary functions may dis- 
appear and be replaced by constants, infinite in number, without the 
integral ceasing to be complete. To elucidate these general considera- 
tions, we apply them to examples by means of which we show that 
the different integrals, in the series of the same equation of partial dif- 
ferences, 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 satis- 
fies 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 @ 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 equa- 
tions 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 solu- 
tion 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 neces- 
sary to proceed in a manner proper to determine their coefficients ac- 
cording to the initial state of the system. On the occasion of a pro- 
blem relative to the heat of a sphere composed of two different sub- — 
stances, I have given for this purpose in the Journal del Ecole Polytech- 
nique, (cahier 19, p.377 et seq.) a direct and general method, of 
