1830.] 
and the Communication of Heat. 
337 
Again, Jet us compare the ratios of cooling, when the surrounding mass is at 40 
and when it is at 60, we shall have 
1,15 1,16 1,16 1,15 1,17 1,16 1,1R 1,16. 
Finally, for the relations of the rates, when the surrounding mass is at 60 and at 
80°, we have 
1,15 1,15 1,16 1,17 1,16 1,17 1,17 1,15. 
The three last comparisons lead us to the same result as the first, and further 
instruct us, that the ratio between any two consecutive series is the same, whether 
the surrounding matter be at 0° and 20° ; at 20 and 40 ; at 40 and 00; or lastly, 
at 60 and 80°. The preceding experiments may then be considered to establish the 
following law : — 
The rote of cooling of a Thermometer in a vacuum, for a certain excess of tempera- 
ture, increases in a geometrical progression , when the temperature of the matter which 
surrounds it, increases in an arithmetical progression. The ratio of this geometric 
progression is the same, whatever the excess of temperature. 
This first law, which only considers a change in the temperature of the surround- 
ing matter, will allow of our putting the formula, previously found for the rate of 
cooling, in a vacuum 
F{t+t)-F{f) 
under the form, 
0 (/) X a* 
a being a constant number, and 0 (/) a function of the variable t singly which it is 
our object to discover. 
The two expressions of the rate of cooling being equal, we have 
F {t + t) - F (t) _ {t) 
which being expanded into series, gives 
F' (t) F" (t) 
F'" (t) 
0 (') = * 
Ac. 
„ 2 2.3 at 
and as this equation must hold good whatever the value of /, it requires, that 
F (t) = m a 1 
m being a number to be determined. Hence we deduce 
F (t) = m a ‘ -f- constant, 
and consequently 
constant. 
Finally, therefore, wo obtain for the velocity of cooling 
V = tn a 1 (a 1 — 1) 
bc s ° ,ikewise> * nd ,he 
.. ^ i ovnrp.ssed as follows : • 
preceding law might be '^^ifthe surrounding matter being kept at a uniform tern - 
]Vhen a body cools m a vac • , tn L era tnre in arithmetical progression 
per at are, the rate of c °° b * 9 f Qmetrica i progression diminished by a constant number. 
ittcreases as the terms of a g „v to determine, in the case of the tliermo- 
The ratio a of this progres «on is™. jjntotorm whmt{ncnaeB by 20% 
meter, the coohng of which as b J to be multiplied by 1,165, being the 
SA we shall .hen have 
„ = 5 V(M65T = 1.0077* 
, nr \. a A olthoue-h we have no intention 
. A coincidence sufficiently corpus to be «“« te ’ nunlber f. , cr y nearly the *)uare 
of drawing any conclusion therefrom ..jtl at tire 
of the coefficient of the expansion ot a gas. 
