of a Mass of Matter. 87 



to the absolute temperature, as is required by the foregoing 

 theorem. 



In the other cases that are quoted, as well as in most others, 

 the relations are less simple, because in them an essential part is 

 played by the forces exerted by the molecules upon one another, 

 forces wnich, as yet, are quite unknown. It results, however, 

 from the mere consideration of the external resistances which 

 heat is capable of overcoming, that in general its force increases 

 with the temperature. If we wish, for instance, to prevent the 

 expansion of a body by means of external pressure, we are obliged 

 to employ a greater pressure the more the body is heated; hence 

 we may conclude, without having a knowledge of the internal 

 forces, that the total amount of the resistances which can be over- 

 come in expansion, increases with the temperature. We cannot, 

 however, directly ascertain whether it increases exactly in the 

 proportion required by the foregoing theorem, without knowing 

 the internal forces. On the other hand, if this theorem be re- 

 garded as proved on other grounds, we may reverse the process, 

 and employ it for the determination of the internal forces exerted 

 by the molecules. 



The forces exerted upon one another by the molecules are not 

 of so simple a kind that each molecule can be replaced by a 

 mere point; for many cases occur in which it can be easily seen 

 that we have not merely to consider the distances of the mole- 

 cules, but also their relative positions. If we take, for example, 

 the melting of ice, there is no doubt that here internal forces, 

 exerted by the molecules upon each other, are overcome, and ac- 

 cordingly disgregation takes place; nevertheless the centres of 

 gravity of the molecules are on the average not so far removed from 

 each other in the liquid water as they were in the ice, for the water 

 is the more dense of the two. Again, the peculiar behaviour of 

 water in contracting when heated above 0° C, and only beginning 

 to expand when its temperature exceeds 4°, shows that likewise 

 in liquid water, in the neighbourhood of its melting-point, 

 increase of disgregation is not connected with increase of the mean 

 distances of its molecules. In the case of the internal forces, it 

 would accordingly be difficult — even if we did not want to mea- 

 sure them, but only to represent them mathematically — to find a 

 fitting expression for them which would admit of a simple deter- 

 mination of magnitude. This difficulty, however, disappears if 

 we take into calculation, not the forces themselves, but the me- 

 chanical work which in any alteration of arrangement is required to 

 overcome them. The expressions for the quantities of work are 

 simpler than those for the corresponding forces; for the quantities 

 of work can be all expressed, without further secondary state- 

 ments, by numbers with the same unit, which can be added 



