258 THEOEY OF HEAT. [CHAP. IV. 



^ +2 _ 2 a . = A sm B + Uj + 2 co* + &c. 

 _ _ 2 a . = A sin (j3 + Uj + 3 ) CD* + &c. 



276. We see, by these equations, that the later differences 

 between the actual temperatures and the final temperatures are 

 represented by the preceding equations, preserving only the first 

 term of the second member of each equation. These later differ 

 ences vary then according to the following law : if we consider 

 only one body, the variable difference in question, that is to say ? 

 the excess of the actual temperature of the body over the final 

 and common temperature, diminishes according to the successive 

 powers of a fraction, as the time increases by equal parts ; and, if 

 we compare at the same instant the temperatures of all the 

 bodies, the difference in question varies proportionally to the suc 

 cessive sines of the circumference divided into equal parts. The 

 temperature of the same body, taken at different successive equal 

 instants, is represented by the ordinates of a logarithmic curve, 

 whose axis is divided into equal parts, and the temperature of 

 each of these bodies, taken at the same instant for all, is repre 

 sented by the ordinates of a circle whose circumference is divided 

 into equal parts. It is easy to see, as we have remarked before, 

 that if the initial temperatures are such, that the differences of 

 these temperatures from the mean or final temperature are pro 

 portional to the successive sines of multiple arcs, these differences 

 will all diminish at the same time without ceasing to be propor 

 tional to the same sines. This law, which governs also the initial 

 temperatures, will not be disturbed by the reciprocal action of the 

 bodies, and will be maintained until they have all acquired a 

 common temperature. The difference will diminish for each body 

 according to the successive powers of the same fraction. Such is 

 the simplest law to which the communication of heat between a 

 succession of equal masses can be submitted. When this law has 

 once been established between the initial temperatures, it is main 

 tained of itself; and when it does not govern the initial tempera 

 tures, that is to say, when the differences of these temperatures 

 from the mean temperature are not proportional to successive 

 sines of multiple arcs, the law in question tends always to be set 



