This equation differs from that of Thomson as given in various places (_e.g._ in the _Encyclopædia Britannica_ article on Elasticity which he also wrote) in the negative sign on the right-hand side, but the difference is only apparent. According to his specification a pressure would be a positive stress, and an expansion a positive displacement, and in applying the equation to numerical examples this must be borne in mind so that the proper signs may be given to each numerical magnitude. As an example of adiabatic change, a sudden extension of the wire already referred to by an increase of stress dS may be considered. If there is not time for the passage of heat from or to the surroundings of the wire, the change of temperature will be given by equation (G). This equation was applied by Thomson (article Elasticity) to find the relation between what he called the kinetic modulus of elasticity and the static modulus, that is, between the modulus for adiabatic strain and the modulus for isothermal strain. The augmentation of the strain produced by raising the temperature 1° is e, and therefore edT, that is, －Te²dS⧸Kρ, is the increase of strain due to the sudden rise of temperature dT. This added to the isothermal strain produced by dS will give the whole adiabatic strain. Thus if M be the static or isothermal modulus, the adiabatic strain is dS⧸M － Te²dS⧸Kρ. If M' denote the kinetic or adiabatic modulus its value is dS divided by the whole adiabatic strain, that is, M' = M⧸(1 － MTe²⧸Kρ) and the ratio M'⧸M = 1⧸(1 － MTe²⧸Kρ). It is well known and easy to prove, without the use of any theorem which can be properly called thermodynamic, that this ratio of moduli is equal to the ratio of the specific heat K of the substance, under the condition of constant stress, to the specific heat N under the condition of constant strain of the corresponding type. This, indeed, is self-evident if two changes of stress, one isothermal the other adiabatic, _which produce the same steps of displacement ds_, be considered, and it be remembered that the step ∂T of temperature which accompanies the adiabatic change may be regarded as made up of a step －dT of temperature, accompanying a displacement ds effected at constant stress, and then two successive steps dT and ∂T effected, at constant strain, along with the steps of stress dS. The ratio M'⧸M is easily seen to have the value (∂T + dT)⧸dt, and since －KdT + N(∂T + dT) = 0, by the adiabatic condition, the theorem is proved. Laplace's celebrated result for air, according to which the adiabatic bulk-modulus is equal to the static bulk-modulus multiplied by the ratio of the specific heat of air pressure constant to the specific heat of air volume constant, is a particular example of this theory.