20 
FIFTH REPORT— 1835 . 
ton’s law to be true, and the conductivities of the substance in 
question to be given, the determination of the progress of heat¬ 
ing and cooling will involve mathematical relations and calcula¬ 
tions, which, as we have seen, had begun to attract attention in 
the middle of the last century. But it was not till the beginning 
of this, that such problems were taken up with due generality, 
and followed into special consequences*. In December 1807 
a memoir of Fourier’s was read at the Institute, which must be 
considered as the commencement of a new mode of treating the 
subject. This memoir was published in 1808 in the .Bulletin 
des Sciences of the Philomathic Society f; and in it was given 
the general partial differential equation between v, the tempera¬ 
ture at a point of any substance of which the coordinates are 
y, s, and t the time; namely, 
dv _ fd~v d 2 v d 2 v \ 
dt \dx 2 dy 2 d z 2 ) 
this, with the equations which belong to the surface and express 
the conditions of exterior conductivity, contains the solution of 
the problem ; though it was only in a few cases, and by means 
of refined analytical artifices, that the integrals of these equations 
could be obtained. 
Problems concerning the motion of heat now drew the atten¬ 
tion of the mathematicians of France; and the solutions of which 
Fourier was known to be in possession awoke the activity of 
other mathematicians. In the Memoirs of the Institute for 1809 
(published in 1810), it is proposed, as the prize-question for 1812 
(p. 96), “To give the mathematical theory of the propagation of 
heat, and to compare this theory with exact observations.” Fou¬ 
rier’s memoir was sent Sept. 28,1811,and the prize was adjudged 
to it, probably as had been expected on all hands, in the ensuing 
January. This memoir, founded upon the one written in 1807, 
crowned in 1812, was not published till 1824, in the Memoirs 
of the Institute for 1819 and 1820; another remarkable example 
of the delay and ambiguity of date in French publications of this 
kind. While Fourier’s memoir thus remained in the archives 
of the Institute, it was consulted by Poisson and Cauchy; in 
the Bulletin des Sciences for 1820 was published an extract 
* “ The form of the equation (the differential equation of the temperature of 
a bar which is come to a permanent state), and the form of the partial differen¬ 
tial equation, which obtains when the bar grows hotter or colder, were indicated 
by M. Biot in 1804, in the extract of a memoir on the propagation of heat (Bi- 
blioth. Britann ., t.xxvii.) M. Biot deduces his equation from Newton’s principle, 
applied to thin contiguous slices, integrates for the permanent condition, and ve¬ 
rifies the result by his own experiments and those of Rumford. (Poisson, Tkeo. 
de la Chal ., 1835, p. 1.) 
f tom. i. p. 112. 
