α In making use of the preceding formula, it is to be observed that the contrary algebraical signs of sin a and sin 3 apply to those cases in which the two angles and 3 lie at contrary sides of O C. In the cases in which those angles lie at the same side of O C, their algebraical signs are the same; and in the formula they are to be made both positive or both negative, according as 3 is less or greater thana; so that the efficiency may be always expressed by a fraction less than unity. That is to say, fr m sin 3 ....(3 A.) 1+ sin s fr 1+ sin a When the lines of action intersect, let O C be denoted by c; then l =c cos a, and m = c cos 8; and consequently the three preceding equations take the following form: When the lines of action of the forces are parallel, we have sin and sin = +1 or 1, as the case may be; and the formulæ take the following shape: When and m lie at contrary sides of O, the piece is a of the first kind;" and "lever When / and m lie at the same side of O; If ml, the piece is a "lever of the second kind;" and If ml, the piece is a "lever of the third kind;" and ..(5 A.) ..(5 B.) (As to levers of the first, second, and third kinds, see Article 221, page 233.) The following method is applicable whether the forces are inclined or parallel; in the former case it is approximate, in the latter exact. Through O, perpendicular to OC, draw ÚO V, cutting the lines of action of the given force and of the effort in U and V respectively. The point where this transverse line cuts the small circle B B coincides exactly with T when the forces are parallel, and is very near T when they are inclined; and in either case the letter T will be used to denote that point. Then It is evident that with a given radius and a given co-efficient of friction, the efficiency of an axle is the greater the more nearly the effort and the given force are brought into direct opposition to each other, and also the more distant their lines of action are from the axis of rotation. 374. Axles of Pulleys connected by Bands.—When the rotating piece which turns with an axle consists of a pair of pulleys, one receiving motion from a driving pulley, and the other communicating motion to a following pulley, regard must be had to the fact that the useful resistance and the driving effort are each of them the difference of a pair of tensions; and that it is upon the resultant of each of those pairs of tensions (being their sum, if they act parallel to each other) that the axle-friction depends. The principles according to which the tensions required at the two sides of a band for transmitting a given effort are determined, have been stated in Article 310 A, pages 351, 352. The belt which drives the first pulley may be called the driving belt; that which is driven by the second pulley, the following belt. The tensions on the two sides of the following belt are given; and the moment of the useful resistance is that of their difference, acting with a leverage equal to the effective radius of the second pulley. Let p be that radius; T1 and T2 the two tensions; then the moment of the useful resistance is PR = p(T1- T2). For the actual useful resistance there is to be substituted a force equal to the resultant of T, and T2, and exerting the same moment. That is to say, let denote the angle which the two sides of the band make with each other; then for the actual useful resistance is to be substituted a force, 1 2 .......... R" = √ {T; + T + 2 T, T2 cos 7}, .(1.) acting at the following perpendicular distance from the axis of rotation: k P (T, T2) (2.) And this is to be compounded with the weight of the rotating piece, to find the given force R' of the rules in the preceding Article, whose perpendicular distance from the axis will be The value of may be expressed in terms of the ratio of the tensions to each other, and independently of their absolute values, T be the ratio of the two tensions found as follows:-Let N = T Ꭲ, by the rules of Article 310 A, page 351. Then k = p (N-1) √ {N2 + 1 + 2 N cos } (4.) In like manner, for the actual line of action of the effort by which the first pulley is driven is to be substituted the line of action of a force exerting the same moment, and equal to the resultant of the tensions of the two sides of the driving-band. The perpendicular distance m of this line of action from the axis of rotation is given by the following formula:-Let p' be the effective radius of the pulley; N', the ratio of the greater to the lesser tension;, the angle which the two sides of the band make with each other; then m = p' (N'-1) 2 (5.) There are many cases in practice in which the two sides of each of the bands may he treated as sensibly parallel; and then we have simply, And if, moreover, as frequently happens, the weight of the pulleys and axle is small compared with the tensions, we may neglect it, and make R' R" and l=k, preparatory to applying the rules of the preceding Article to the determination of the efficiency. 375. Efficiency of a Screw.-The efficiency of a screw acting as a primary piece is nearly the same with that of a block sliding on a straight guide, which represents the development of a helix situated midway between the outer and inner edges of the screw-thread ; the block being acted upon by forces making the same angles with the straight guide that the actual forces do with that helix. As to the development of a helix, see Article 63, page 40; and as to the efficiency of a piece sliding along a straight guide, see Article 372, page 426. 376. Efficiency of Long Lines of Horizontal Shafting.-In a line of horizontal shafting for transmitting motive power to long distances in a mill, a great part of the wastel work is spent in overcoming the friction produced simply by the weight of the shaft resting on its bearings; and the efficiency and counterefficiency as affected by this cause of loss of power can be considered and calculated separately. For reasons connected with the principles of the strength of materials, to be explained further on, the cube of the diameter of a shaft of uniform diameter must be made to bear a certain proportion to the driving moment exerted upon it to keep up its rotation. That is to say, let M, denote that moment; h, the diameter of the shaft; then A being a co-efficient whose values in practice range, according to circumstances to be explained in the Third Part of this treatise, for forces in lbs. and dimensions in inches, from 300 to 1,800; and for forces in kilogrammes and dimensions in millimètres, from 0.21 to 1.26. Let w denote the heaviness of iron; f, the co-efficient of friction; then the weight of an unit of length of the shaft is the friction per unit of length is, very nearly, and the moment of friction per unit of length is f wh3 33927 fw h3 nearly.......(2) Let L be the length of a shaft of uniform diameter, such that the whole driving moment is exhausted in overcoming its own friction. This may be called the exhaustive length. Then we must have M1 = A 33927 fw h3 L; and therefore 10 For lengths in feet, and diameter in inches, we have w= 3 ; being the weight in pounds of a rod of iron a foot long and an inch square. For lengths in mètres, and diameters in millimètres, we have w = 0077 nearly; being the weight of a rod of iron one mètre long and one millimètre square. Let f = 0·051; then the following are the values of the exhaustive length L corresponding to different values of A: It is obvious that the efficiency of a length, l, of shafting of uniform diameter is given by the expression Mo =1 M1 1 Mo being the driving moment in the absence of friction; M1, the L' actual driving moment; and the fraction of that moment expended on friction; also, that the counter-efficiency is When, besides its own weight, the shaft is loaded with the weights of pulleys and tensions of belts, the effect of such additional load |