The ingenious method by which the weight of the platform is eliminated from the result of the work of the machine was exhibited and explained. This is accomplished by counterweights sliding automatically in tubes, so that in any position the unloaded platform is always in equilibrium. Any combination of representative weights can then be placed on this platform at the proper points of the scale. By then drawing the platform to its balancing point, the location of the center of gravity will at once be indicated on the scale by the pointer over the central trunnion. The weights may be arranged on a decimal system, with intermediate weights for closer working, or they may be made so as to express multiples or factors. Each machine is provided with a number of differing scales, divided suitably for various purposes. When the problem is one of time, the scale represents months and days; for problems of proportion, the zero of the scale is at the center of its length; for problems for the location of center of gravity of a system from a fixed point, the zero is at the extremity of the scale, etc. The machine exhibited has sixty-three transverse grooves, which, by arrangement of weights, can be made to serve the purposes of two hundred and fifty-two grooves. The machine is 29 inches in length, 9 inches in width, and weighs about 13 pounds. With the machine can be found average dates, as, for instance, of purchases and of payments extending over irregular periods; also average prices, as for "futures," in comman use among cotton brokers. The problem of average haul, so often presented to the engineer, can be solved with ease and great celerity. Practical examples of the solution of these and a number of other problems involving proportions or averages were given by the author. * * * * * COMPOUND BEAM ENGINE. The engine represented in Figs. 1 to 4 herewith is intended for a mill, and is of 530 to 800 indicated horse-power, the pressure being seven atmospheres, and the number of revolutions forty-five per minute. As will be seen by the drawing each cylinder is placed in a separate foundation plate, the two connecting rods acting upon cranks keyed at right angles upon the shaft, W, which carries the drum, T. The high-pressure cylinder, C, is 760 mm diameter, the low pressure cylinder being 1,220 mm. diameter, and the piston speed 2.28 m. The drum, which also fulfills the purpose of a fly wheel, is provided with twenty-eight grooves for ropes of 50 mm. diameter. With the exception of the cylinders, pistons, valves, and valve chests, the engines are of the same size, corresponding to the equal maximum pressures which come into action in each cylinder, and in this respect alone the engine differs in principle from an ordinary twin machine. [Illustration: BORSIG'S IMPROVED COPOUND BEAM ENGINE. FIG. 1]