REPORT ON ELECTRICITY, MAGNETISM, AND HEAT. 
25 
be discontinuous , and we are thus led to that curious and per¬ 
plexing part of analysis which treats of such functions. It is 
the less necessary for me to dwell on this train of investigation, 
in as much as it has been fully treated of by Mr. Peacock in 
his Report on the Progress of Analysis, read last year*. I will 
only observe, that the general solution f is in this case expressed 
by an infinite series of such terms as I have mentioned, the series 
having such coefficients that, at the extremity of the bar, the 
function represented by it is discontinuous. 
In the case of the lamina just spoken of, we have two dimen¬ 
sions of space to consider (length and breadth, the thickness 
being left out of view); but we have nothing to do with the time, 
because we consider only the ultimate and permanent condition 
of the body. Another of the problems treated by Fourier, that of 
the distribution of heat in a sphere, is simplified in a different 
way. By supposing the heat to be uniformly distributed about 
the centre, the temperature at any point depends only upon the 
distance from the centre and the time ; and the differential 
equation which expresses this dependence may be obtained£. 
But in this case we have necessarily a second differential equa¬ 
tion §, which expresses the conditions of radiation at the surface, 
as the first expresses the conditions of conduction in the interior. 
In this case also we can easily assign a particular solution ||; 
or a possible distribution of the heat and a possible relation of 
the conductive and radiative powers, which shall cause the 
cooling of the sphere to follow a certain law with respect to the 
times; namely, that the temperature shall diminish in geome¬ 
trical progression for equal increments of time. The extending 
this particular into a general solution consists, in this as in the 
former case, in adding together a number of simple terms of 
this kind, and in so determining them that they shall agree w T ith 
the given relation of conductive and radiative powers of the 
globed, and also that the original distribution of heat shall be 
* Report of Third Meeting, p. 251, et seq. 
f —' = s * cosy—-s 3x cos 3 y -f- -e 5 ‘* cos 5 y — &c.— -Theorie de la 
1 4 J 3 5 
Ckaleur, p. 190. 
+ dv _ 
d t 
dv 
= lc(pL+lf\ 
\dx z x dx/ 
§ -h h v = 0 when x — X, X being the whole radius. 
dx 
. sin (n x) -lent 
= A —^' g 
x 
^nX = (l-\x) tan n X, which determines n. 
