237. The expenditure of work due to the friction of the axes. This expenditure is represented by the formula forming the second term of formula 280. Now, evidently, the value of this formula is less as the quantities sin. 1, sin. 2, &c. are less, or as the limiting angles of resistance between the surfaces of the axes and their bearings are less, or the lubrication of the axes more perfect; and it is less as the LIP1 Lapa L3P3 fractions &c. are less. Now, L, being the distance between the point of contact b of the third and fourth wheels and the projection of the point of contact a of the first and second upon the plane of those wheels, it follows that, generally, L. is least when the projection of a falls on the same side of the axis as the point b*; and that it is least of all when this line falls on that side and in the line joining the axis with the point b; whilst it is greatest of all when it falls in this line produced to the opposite side of the axis. In the former case its value is represented by r2-r and in the latter by r+r; so that, generally, the maximum and minimum values of L, are represented by the expression rar, and the maximum and minimum values of by P2. And similarly it appears that the maximum and 1 L3P3 minimum values of r1rs are represented by (+) Pai and so of the rest. So that the maximum and minimum values of * This important condition is but a particular case of the general principle established in Art. 168.; from which principle it follows, that the driving pressure on each wheel should be applied on the same side of the axis as the driven pressure. the work lost by the friction of the axes are represented by the expression from which expression it is manifest, that in every case the expenditure of work due to the friction of the axes is less as the radii of the axes are less when compared with the radii of the wheels; being wholly independent of actual dimensions of these radii, but only upon the ratio or proportion of the radius of each axis to that of its corresponding wheel: moreover, that this expenditure of work is the least when the wheels of the train are so arranged, that the projection of the point of contact of any pair upon the plane of the next following pair shall lie in the line of centres of this last pair, between their point of contact and the axis of the driving wheel of the pair; whilst the expenditure is greatest when this projection falls in that line but on the other side of the axis. The difference of the expenditures of work on the friction of the axes under these two different arrangements of the train is represented by the formula which, in a train of a great number of wheels, may amount to a considerable fraction of U,; that fraction of U, representing the amount of power which may be sacrificed by a false arrangement of the points of contact of the wheels. 238. The expenditure of work due to the weights of the several wheels of the train. The third and last term N. S of the expression (280) represents the expenditure of work due to the weights of the several wheels of the train; of this term the factor N is represented by an expression (equation 279), each of the terms of which involves as a factor one of the quantities N,, N2, N3, &c. whose general type or form is that given in equation (247), it being observed that the direction of the driving pressure on any pair of the wheels being supposed that of a tangent to their point of contact; the case is that discussed in the note to page 292. The other factor of each term of the expression (equation 279) for N, is a fraction having the product n, ng... of the numbers of teeth in all the preceding drivers of the train, except the first, for its numerator, and the product n,. n. n... of the numbers of teeth in the preceding followers of the train for its denominator; so that if the train be one by which the motion is to be accelerated, the numbers of teeth in the followers being small as compared with those in the drivers, or if the multiplying power of the train be great, and if the quantities N1, N2, N3, &c. be all positive; then is the expenditure of work by reason of the weights of the wheels considerable, as compared with the whole expenditure. Since, moreover, the coefficients of N1, N2, N3, &c. in the expression for N (equation 279) increase rapidly in value, this expenditure of work is the greatest in respect to those wheels of the train which are farthest removed from its first driving wheel: for which reason, especially, it is advisable to diminish the weights of the wheels as they recede from the driving point of the train, which may readily be done, since the strain upon each successive wheel is less, as the work is transferred to it under a more rapid motion. 239. The modulus of a train in which all the drivers are equal to one another and all the followers, and in which the points of contact of the drivers and followers are all similarly situated. = The numbers of teeth in the drivers of the train being in this case supposed equal, and also the radii of these wheels, n1 =n=n=n,= &c., r1=r3=r=r, &c. The numbers of teeth in the followers being also equal, and also the radii of the followers n=n ̧=n=&c., r2=r1=r«=&c. = If moreover, to simplify the investigation, the driving work U1 be supposed to be done upon the first wheel of the train 9 at a point situated in respect to the point of contact a of that wheel with its pinion precisely as that point of contact is in respect to the point of contact b of the next pair of wheels of the train; and if a similar supposition be made in respect to the point at which the driven work U, is done upon the last pinion of the train, then, evidently, L1=L=L2=...=L„, and (see equation 247) N1 = N,=...=N. The modulus (equation 280) will become, these substitutions being made in it, the axes being, moreover, supposed all to be of the same dimensions and material, and equally lubricated, and it being observed that the drivers and the followers are each p in number, 1 1 U1 = { 1 + xp (+) sin. + P which is the modulus required. Lisin.U,+NS.. 2 Moreover, the value of N (equation 277) will become by THE TRAIN OF LEAST RESISTANCE. 240. A train of equal driving wheels and equal followers being required to yield at the last wheel of the train a given amount of work U2, under a velocity m times greater or less than that under which the work U, which drives the train is done by the moving power upon the first wheel; it is required to determine what should be the number p of pairs of wheels in the train, so that the work U1 ex- Since the number of revolutions made by the last wheel of Substituting these values in the modulus (equation 284); U, = {1+2,p(m2 + 1) sin. p + (-;;?;2)pm2 sin., U+N, (m-1) (m2)s m"-1 S. (286). It is evident that the question is solved by that value of p which renders this function a minimum, or which satisfies the dU, d2U, conditions 1-0 and >0. The first condition gives dp dp2 This equation may be solved in respect to p, for any given values of the other quantities which enter into it, by approximation. If, being differentiated a second time, the above expression represents a positive quantity when the value of p (before determined) is substituted in it, then does that value satisfy both the conditions of a minimum, and supplies, therefore, its solution to the problem. Ꮓ |