corresponding to Fig. 37, or they may fall on opposite sides of the belt line as indicated in Fig. 43. Although the appearance of the diagram is quite different in each of the three cases, the principle and process are the same. 11. Having drawn the first belt line HI, as in rules 1 and 2, proceed as follows: If the angle of HI, Figs. 41, 42 and 43, is less than 18° describe the arc TWS tangent to it from the center G. Now draw a straight line, SU, tangent to this arc, WS, making an angle of 18° with the line of centers, EF. Tangent to this line, SU, draw another arc, UX, from the center m. These two arcs, WS and UX, constitute the directing curve and may be duplicated on the opposite side of BG, WT and YV, if desirable, but is not necessary. If the first belt angle is greater than 18°, then the process is reversed, that is, the first arc is drawn tangent to the belt line from the center m instead of G; the line SU drawn tangent to this arc at an angle of 18°; and the second arc, SW, is drawn tangent to this line from the center G. The principle is, that all belt lines with an angle smaller than 18° are tangent to the arc WS, from the center G, while for all belt angles greater than 18° the belt line should be tangent to the arc UX, whose center is at m. The complete construction is shown in Fig. 44, which is drawn to scale from example 1, page 291. The step pulleys corresponding to this example are shown in Fig. 53 and the cone pulleys in Fig. 54. The method seems much more complicated from the description than it really is, and it takes a great deal more space and time to explain than to practice it. The work of laying out the diagram and finding the diameters is really quite simple when the different steps are once fixed in the mind. GRAPHICAL METHOD FOR CONE PULLEYS WITH CROSSED BELT. The solution for crossed belts is usually very simple by the 12 D3 Fig. 46. arithmetical process, but the following graphical method is very convenient. 1. Having determined the first pair of diameters, D, and d1, from known conditions, lay off the line AB, Fig. 45, equal to the sum of these diameters; that is, AH being equal to the diameter of the large pulley and HB the small one. 2. Draw the line CB at right angles to AB and erect the perpendicular EG at any convenient distance, GB = a, from B. = 3. Lay off the point C, the distance CG b being obtained by multiplying the distance a by the desired velocity ratio, r, of the next pair of pulleys. 4. Connect A and C by a straight line. This will determine the intersection E. The lines GE= BF and FA are the diameters of the pulleys sought, having the desired ratio, r, and will work properly with the first pair, BH and HA. steps may thus be determined. The line CA drawn from the same point, A, wherever the location of the point C' may be. Any number of must always be Fig. 46 shows the complete solution of a pair of cone pulleys, for crossed belt, shown in Fig. 47, D, and d1 being each 10" and the velocity ratios, 1, 2, 3 and 4. MATHEMATICAL METHOD FOR CONE PULLEYS WITH OPEN BELT. The following is an explanation of the symbols used in the succeeding formulæ and throughout this paper. When the letters are used without a subscript, they stand for the quantity in general. When the subscript, 1, 2, 3, etc., is attached to the letter it stands for the first, second, third, etc., quantity for a pair of cones represented by that letter. = A The angle, in degrees, between the center line and the belt of any pair of pulleys. a = .314 for belt angles less than 18° and .298 for belt angles between 18° and 30°. (See page 286.) B = An angle depending upon the velocity ratio. See equation 3 and Fig. 50. C=The center distance of the two pulleys. "smaller of a pair of pulleys. E° An angle depending upon B° and some constants. See equation 4 and Fig. 50. L= The length of the belt when drawn tight around the pulleys. 50. D r = ; or the velocity ratio (the larger divided by the smaller). d R = A constant (radius of directrix). See equation 2 and Fig. Ꭱ 3. . aC - D1, when A1 = 0 and r1 = 1. (d) R2 = R1 (b) d=0.3183 (L−2C), when A=0 and r-1. In calculating the diameters of any pair of cones, equations 1 and 2 need be solved but once, as R1 or R is constant and is used in equation 4 for calculating all the steps. From equation 1 we can obtain the belt angle of any pair of pulleys, which, substituted in 2 (a), gives the value of R to be used in equation 4. In all equations containing the quantity a, make a = .314 when A is less than 18°, and a = .298 when A lies between 18° and 30°. For R, in equation 4, use the value of R, as obtained from |