only one-twelfth, or a little more than 8 per cent. of the total heat is converted; so that whether we look at the total quantity of heat which we show unconverted, or the total heat converted, we find that each supplements and corroborates the other.  If we take the efficiency of the engine alone, without considering the loss caused by the boiler, we find that the 25,168[theta] which entered the boiler should have given 19,429,696 foot-pounds; so that the 2 millions given by the engine represent about 10 per cent. of the heat which has left the boiler.  The foregoing figures refer to large stationary or marine engines, with first-rate boilers.  When, however, we come to high-pressure engines of the best type, the consumption of coal is twice as much; and for those of any ordinary type it is usual to calculate 1 cubic foot, or 621/2 lb., of water evaporated per horse power.  This would reduce the efficiency to about 6 per cent. for the best, and 3 per cent. for the ordinary non-condensing engines; and if to this we add the inefficiency of some boilers, it is certain that many small engines do not convert into power more than 2 per cent. of the potential energy contained in the coal.

At one time the steam-engine was threatened with serious rivalry by the hot-air engine.  About the year 1816 the Rev. Mr. Stirling, a Scotch clergyman, invented one which a member of this Institute (Mr. George Anderson) remembers to have seen still at work at Dundee.  The principle of it was that a quantity of air under pressure was moved by a mass, called a “displacer,” from the cold to the hot end of a large vessel which was heated by a fire beneath and cooled by a current of water above.  The same air was alternately heated and cooled, expanded and contracted; and by the difference of pressure moved the piston in a working cylinder.  In this arrangement the furnace was inefficient.  As only a small portion of heat reached the compressed air, the loss by radiation was very great, and the wear and tear exceedingly heavy.  This system, with some modifications, was revived by Rankine, Ericsson, Laubereau, Ryder, Buckett, and Bailey.  Siemens employed a similar system, only substituting steam for air.  Another system, originally proposed by Sir George Cayley, consisted in compressing by a pump cold air which was subsequently passed partly through a furnace, and, expanding, moved a larger piston at the same pressure; and the difference of the areas of the pistons multiplied by the pressure common to both represented the indicated power.  This principle was subsequently developed by a very able mechanic, Mr. Wenham; but his engine never came much into favor.  The only hot-air engines at present in use are Ryder’s, Buckett’s, and Bailey’s, employed to a limited extent for small powers.  I have not said anything of the thermal principles involved in the construction of these engines, as they are precisely the same as those affecting the subject of the present paper.