1819.], and on the Laws of the Communication of Heat. 335 
The quantities a and 6 will be for all bodies and in all fluids 
equal, the first to 1-0077, and the second to 1-233. The coeffi- 
cient m will depend on the size and the nature of the surface, as 
well as upon the absolute nature of the surrounding body. ‘The 
coefficient n, independent of this absolute temperature, as well 
as of the nature of the surface of the body, will vary with the 
elasticity and with the nature of the gas in which the body is 
plunged; and these variations will follow laws which we have 
already established. 
This formula shows us in the first place, as we have announced 
at the commencement of this memor, that the law of cooling in 
elastic fluids changes with the nature of the surface of the body. 
In fact, when this change takes place, the quantities a, b, and n, 
preserve their values ; but the coefficient m varies proportionally 
to the radiating power of the surface. If we represent its new 
value by m’, the velocity of cooling will become : 
m {a —1l)+ni®; 
a quantity which does not remain proportional to 
m(a— lb + nt', 
when ¢ changes. 
Let us now examine how the ratio of these two velocities 
varies, and let us suppose, in order to fix our ideas, that m is. 
greater than m’; that is to say, that it belongs to the body which 
radiates most. 
We may in the first place satisfy ourselves by means of the 
rules of the differential calculus, that the fraction 
m(at—1)+ntb 
m'(at—1l)+nt, 
becomes equal to ” » whether we make ¢ = 0, orf = wo, 
© If we suppose ¢ very small, the quantity a‘ — 1 is reduced to 
é. log. a, and the preceding ratio becomes, dividing by ¢ log. a, 
n 
iL) 
es log. a 
n 
a! + 
log. a 
- Under this form it is evident that the ratio must diminish in 
proportion as ¢ increases, b being greater than 1 ; but this ratio, 
after having diminished, will again increase, since it must resume 
to infinity the value which it has when ¢ = 0. From this it is 
easy to conclude the truth of the principle which we have esta- 
blished at the beginning of this memoir, and which comes to 
this, that when we compare the laws of cooling in two bodies 
with different surfaces, the law is more rapid at low temperatures 
for the body which radiates the least; and less rapid, on the 
contrary, for the same body, at high temperatures. 
This may be easily verified in the following table, where we 
have inserted the velocities of cooling of the naked thermometer, 
BA aba 
; t°-*! 
