One of the most economical freight-carrying roads in the United States is the Lake Shore and Michigan Southern of the New York Central system, running from Buffalo to Chicago. Its maximum grade is 0.4 of 1 per cent. The maximum grade of the N. Y. C. & H. R. R. R. is 0.75 of 1 per cent between New York City and Albany and between Albany and Buffalo, 1.74 per cent at Albany, 1.12 per cent at Schenectady, and 1 per cent at Batavia. Pushers or assistant locomotives are used for heavy trains at the three latter points. The maximum grade of the Pennsylvania R. R. on the famous Horseshoe Curve between Altoona and Cresson is 1.8 per cent. It is advantageous, whereever practicable, to concentrate heavy grades within a short distance, as in the case of the New York Central at Albany, and use auxiliary engines, called pushers or assistants. Some of the heaviest grades used in this country are found on the trans-continental lines where they pass the summits of the Rocky Mountains or the Sierras. In one portion of its line over a stretch of 25.4 miles the Southern Pacific R. R. rises 2674 feet with a maximum grade of 2.2 per cent; also approaching the Tehacipi Pass in California the maximum grade is about 2.4 per cent. At the Marshall Pass on the Denver & Rio Grande R. R. there is a rise of 3675 feet in 25 miles with a maximum grade of 4 per cent. The Central Pacific R. R. (now a part of the Southern Pacific system) rises 992 feet in 13 miles with a maximum grade of 2 per cent. The Northern Pacific R. R. rises at one place 1668 feet in an air-line distance of 13 miles with a maximum grade of 2.2 per cent. Probably the heaviest grade in the world on an ordinary steam railroad is that of the Calumet Mine branch of the Denver & Rio Grande R. R., which makes an elevation of 2700 feet in 7 miles on an 8 per cent grade and with 25° curves as maximum curvature. These instances are sufficient to illustrate maximum railroad grades found in the United States. =257. Curves.=—Civil engineers in different parts of the world have rather peculiar classifications of curves. In this country the railroad curve is indicated by the number of degrees in it which subtend a chord 100 feet in length. Evidently the smaller the radius or the sharper the curvature the greater will be the number of degrees between the radii drawn from the centre of a circle to the extremities of a 100-feet chord. American civil engineers use this system for the reason that the usual tape or chain used in railroad surveying is 100 feet long. A very simple and elementary trigonometric analysis shows that under this system the radius of any curve will be equal to 50 divided by the sine of one half of the angle between the two radii drawn to the extremities of the 100-feet chord. In other words, it is equal to 50 divided by the sine of one half the degree of curvature. The application of this simple formula will give the following tabular values of the radii for the curves indicated: