After it had been proved that heat must _disappear_ if mechanical work was to be done at its expense, Carnot's principle could no longer be regarded as a complete expression of the facts. Its improved form was first given, in 1850, by Clausius, whom Thomson followed in 1851. It runs thus: "If a quantity of heat _Q'_ is transformed into work in a reversible process, _another_ quantity of heat _Q_ of the absolute[55] temperature _T₁_ is lowered to the absolute temperature _T₂_." Here _Q'_ is dependent only on _Q_, _T₁_, _T₂_, but is independent of the substances used and of the character of the process, so far as that is unaccompanied by loss. Owing to this last fact, it is sufficient to find the relation which obtains for some one well-known physical substance, say a gas, and some definite simple process. The relation found will be the one that holds generally. We get, thus, _Q'/(Q' + Q) = (T₁-T₂)/T₁_ (1) that is, the quotient of the available heat _Q'_ transformed into work divided by the sum of the transformed and transferred heats (the total sum used), the so-called _economical coefficient_ of the process, is, _(T₁-T₂)/T₁_. IV. THE CONCEPTIONS OF HEAT. When a cold body is put in contact with a warm body it is observed that the first body is warmed and that the second body is cooled. We may say that the first body is warmed _at the expense of_ the second body. This suggests the notion of a thing, or heat-substance, which passes from the one body to the other. If two masses of water _m_, _m'_, of unequal temperatures, be put together, it will be found, upon the rapid equalisation of the temperatures, that the respective changes of temperatures _u_ and _u'_ are inversely proportional to the masses and of opposite signs, so that the algebraical sum of the products is, _mu + m'u' = 0_. Black called the products _mu_, _m'u'_, which are decisive for our knowledge of the process, _quantities of heat_. We may form a very clear _picture_ of these products by conceiving them with Black as measures of the quantities of some substance. But the essential thing is not this picture but the _constancy_ of the sum of these products in simple processes of conduction. If a quantity of heat disappears at one point, an equally large quantity will make its appearance at some other point. The retention of this idea leads to the discovery of specific heat. Black, finally, perceives that also something else may appear for a vanished quantity of heat, namely: the fusion or vaporisation of a definite quantity of matter. He adheres here still to this favorite view, though with some freedom, and considers the vanished quantity of heat as still present, but as _latent_.