of Edinburgh, Session 1867-68. 
309 
3. Physical Proof that the Geometric Mean of any Number 
of Quantities is less than the Arithmetic Mean. By 
Professor Tait. 
If a number of equal masses of the same material be given, at 
different temperatures, and enclosed in an envelope impervious to 
heat, they will finally assume a common temperature ; which is the 
arithmetic mean of the initial temperatures, if the material be one 
whose specific heat does not vary with temperature. 
But they may be brought to a common temperature by means of 
reversible thermodynamic engines employed to obtain the utmost 
amount of work from the initial unequal distribution. This ques- 
tion was first investigated by Thomson ( Phil . Mag. 1853, u On the 
Restoration of Energy from an unequally heated Space”), and the 
application of his method to the present problem shows that the 
final common temperature of the masses, when as much work as 
possible has been obtained from them, is the geometric mean of 
the initial temperatures ; but this investigation introduces the con- 
dition that the temperatures must be measured from the absolute 
zero. 
Obviously the whole energy restored is proportional to the ex- 
cess of the arithmetic over the geometric mean. 
Far more complex analytical theorems may easily be proved 
by means of the above process; for instance, if t l} t 2 , , 
c 1? c 2 . . . . be any positive quantities, we have 
C l + c 2 1 2 + 
+ c 2 + . 
— > ft ' 1 ^ * • * * K+C 2+ ... 
4. On the Dissipation of Energy. By Professor Tait. 
The paper contains some curious applications of the principle of 
dissipation to the conduction of heat, the connection of heat and 
electricity, thermo-electric currents, the electric convection of heat, 
&c. But in this abstract we confine ourselves to one very simple 
case of the conduction of heat, as the hypothesis on which it is in 
vestigated is fundamentally assumed in all the other applications. 
