Moreover, the value of N (equation 279) will become by the like substitutions,

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 U,, 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 U, expended through a given space S, in driving it, may be a minimum.

Since the number of revolutions made by the last wheel

of the train is required to be a given multiple or part of the number of revolutions made by the first wheel, which multiple or part is represented by m, therefore (equation 231),

Substituting these values in the modulus (equation 284); substituting, moreover, for N its value from equation (285), we have

1

U,= {1+ = p(m3 +1) sin.❤+ (4.0) poursin. F.3

d'U,
dp

It is evident that the question is solved by that value of p which renders this function a minimum, or which satisfies

the conditions =0 and

dU, dp

gives by the differentiation of equation (286),

> 0. The first condition

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.

If we suppose,=0 and N,=0, or, in other words, if we neglect the influence of the friction of the axes and of the weights of the wheels of the train upon the conditions of the question, we shall obtain

This formula was given by the late Mr. Davis Gilbert, in his paper on the "Progressive improvements made in the efficiency of steam engines in Cornwall," published in the Transactions of the Royal Society for 1830. Towards the conclusion of that paper, Mr. Gilbert has treated of the methods best adapted for imparting great angular velocities, and, in connection with that subject, of the friction of toothed wheels; having reference to the friction of the surfaces of their teeth alone, and neglecting all consideration of the influ

THE INCLINED PLANE.

241. Let AB represent the surface of an inclined plane on which is supported a body whose centre of gravity is C, and its weight W, by means of a pressure acting in any direction, and which may be supposed to be supplied by the tension of a cord passing over a pulley and carrying at its extremity a weight.

Let OR represent the direction of the resultant of P and W. If the direction of this line be inclined to the perpendicular ST to the surface of the plane, at an angle OST equal to the limiting angle of resistance, on that side of ST which is farthest from the summit B of the plane (as in fig. 1), the body will be upon the point of slipping upwards; and if it be inclined to the perpendicular at an angle OST,

The

ence due to the weights of the wheels and to the friction of their axes. author has in vain endeavoured to follow out the condensed reasoning by which Mr. Gilbert has arrived at this remarkable result; it supplies another example of that rare sagacity which he was accustomed to bring to the discussion of questions of practical science. Mr. Gilbert has given the following examples of the solution of the formula by the method of approximation:-If m=120, or if the velocity is to be increased by the train 120 times, then the value of p given by the above formula, or the number of pairs of wheels which should compose the train, so that it may work with a minimum resistance, reference being had only to the friction of the surfaces of the teeth, is 3.745; and the value

1

the factor p(mP+1) (equation 286), which being multiplied by

1 presents the work expended on the friction of the surfaces of the teeth, is in this case 179; whereas its value would, according to Mr. Gilbert, be 121 if the velocity were got up by a single pair of wheels. So that the work lost by the friction of the teeth in the one case would only be one seventh part of that in the other. In like manner Mr. Gilbert found, that if m=100, then p should equal 3.6; in which case the loss by friction of the teeth would amount to the sixth part only of the loss that would result from that cause if p=1, or if the required velocity were got up by one pair of wheels.

If m=40, then p=2.88, with a gain of one third over a single pair.

If m=3.59, then p=1.

If m=12.85, then p=2.

If m=463, then p=3.

If m=166-4, then p=4.

It is evident that when p, in any of the above examples, appears under the form of a fraction, the nearest whole number to it, must be taken in practice. The influence of the weights of the wheels of the train, and that of the friction of the axes, so greatly however modify these results, that although they are fully sufficient to show the existence in every case of a certain number of wheels, which being assigned to a train destined to produce a given accelera. tion of motion shall cause that train to produce the required effect with the least expenditure of power, yet they do not in any case determine correctly what that number of wheels should be.

equal to the limiting angle of resistance, but on the side of ST nearest to the summit B (as in fig. 2.), then the body will be upon the point of slipping downwards (Art. 138.); the former condition corresponds to the superior and the latter to the inferior state bordering upon motion (Art. 140.).

Now the resistance of the plane is equal and opposite to the resultant of P and W; let it be represented by R. There are then three pressures P, W, and R in equilibrium.

Let BAC=1, OST=lims. of resistance=9, let represent the inclination PQB of the direction of P to the surface of the plane, and draw OV perpendicular to AB; then,

in fig. 1, WOR-WOV+SOV=BAC+OST=1+9, and POR=PQB+OSQ=PQB+~—OST="~+8—9; in fig. 2., WOR-WOV-SOV=BAC-OST=1-0, and POR=PQB+O6Q=PQB++OST=++;

the

2

2

upper or lower sign being taken according as the body is upon the point of sliding up the plane, as in fig. 1, or down the plane, as in fig. 2. Or if we suppose the angle to be taken positively or negatively according as the body is on the point of slipping upwards or downwards; then generally WOR=1+9 POR=2+(6−q);

If the direction of P be parallel to the plane, PQB or 9=0; and the above relation becomes

If 0 the plane becomes horizontal (fig. 3)., and the relation between P and W assumes the form

If 0=0, P=W. tan. o, as it ought (see Art. 138.).

If the angle PQB or (fig. 1.) be increased so as to become, PQ will assume the direction shown in fig. 4, and the relation (equation 289), between P and W will be

The negative sign showing that the direction of P must, in order that the body may slip up the plane, be opposite to that assumed in fig. 1.; or that it must be a pushing pressure in the direction PO instead of a pulling pressure in the direction OP.

If, however, the body be upon the point of slipping down the plane, so that must be taken negatively; and if, moreover, be greater than, then sin. (+), will become sin. (—)——sin. (p-1), so that P will in this case assume the positive value

P=W.

sin. (-1)

cos. (-)

(293),