But we can also cause heat to flow from a colder to a hotter body _by effecting a compensatory energy-transformation_. Such a compensation would not occur by itself in any system capable of effecting an energy-transformation, if it is to be effected some external agency must act on the transforming system. We can suppose it to happen in a perfectly reversible imaginary mechanism. Suppose a Carnot engine works in the positive direction, taking heat from a reservoir at temperature _T_↓{2}°, and giving up part of this heat to a refrigerator at _T_↓{1}°, and doing a certain amount of work _W_. Suppose that this work is stored up, so to speak, say by raising a heavy weight, which can then fall and actuate the same Carnot engine in the opposite (negative) direction. The engine then exactly reverses its former series of operations. The work it did is reconverted into heat, and as much of this heat flows from the refrigerator into the source, that is, from a colder to a hotter body, in the negative operations, as flowed from the source to the refrigerator in the positive operations. In this primary energy-transformation, combined with a compensatory energy-transformation, there is no change of entropy. The mechanism is an ideal one--the limit to an irreversible mechanism. But--and now we appeal to experience and cease to work with ideal mechanisms--the actual engine which we can design and work is one in which there will be friction, in which some parts will conduct heat imperfectly, and other parts will insulate heat imperfectly. Let the friction generate _q_ units of heat, and let the quantity of heat which is “wasted” by imperfect conduction and insulation be _q_↓{1}. This heat will flow into the refrigerator, or will be radiated or conducted to the surrounding medium, which we suppose to be at the same temperature as the refrigerator. If, then, we divide this total quantity of heat by the temperature _T_↓{1}°, we get (_q_ + _q_↓{1})/_T_↓{1}° = _S_↓{1} as the quantity of entropy which is generated as the result of the imperfections of the engine, in addition to the quantity of entropy, _S_, which would be generated if the engine were a perfect one. Both _S_ and _S_↓{1}_ are positive. Also in the working of the engine in the negative direction a certain quantity of entropy, _S_↓{1}, is generated for reasons similar to those mentioned above. The entropy generated when the engine works in the positive direction is therefore _S_ + _S_↓{1}, and when it works negatively the quantity generated is also _S_↓{1}. The entropy destroyed when the engine works negatively is _S_. The total change of entropy is therefore 2_S_↓{1} + _S_ - _S_, that is, 2_S_↓{1}. In an actual energy-transformation combined with a compensatory energy-transformation there is therefore an increase of entropy.