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wheel of the same weight, having a radius equal to the mean radius of the conical wheel, and revolving also in a vertical plane. If the axis of the wheel be not horizontal, its weight must be resolved into two pressures, one acting in the plane of the wheel, and the other at right angles to it; the latter is effective only on the extremity of the axis, where it is borne as by a pivot, so that the work expended by reason of it may be determined by Art. 176, and will be found to present itself under the form of N, . S, where N, is a constant and S the space described by the pitch circle of the wheel, whilst the work U, is done. The resolved weight in the plane of the wheel must be substituted for the weight of the wheel in equation (247), which determines the value of N. Assuming the value of N, this substitution being made, to be repre sented by N,, the whole of the second term of the modulus will thus present itself under the form (N,+N,)S.

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233. Comparing the modulus of a system of two conical wheels with that of a system of two cylindrical wheels (equation 252), it will be seen that the fractional excess of the work U, lost by the friction of the latter over that lost by the friction of the former is represented by the formula

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The first term of this expression is due to the friction of the teeth of the wheels alone, as distinguished from the friction of their axes; the latter is due exclusively to the friction of the axes. Both terms are essentially positive, since 7. every case less than

and

72 are in

2

Thus, then, it appears that the loss of power due to the friction of bevil wheels is (other things being the same) essentially less than that due to the friction of spur wheels, so that there is an economy of power in the substitution of

a bevil for a spur wheel wherever such substitution is prac ticable. This result is entirely consistent with the experience of engineers, to whom it is well known that bevil wheels run lighter than spur wheels.

234. THE MODULUS OF A TRAIN OF WHEELS.

In a train of wheels such as that shown in the accompany

ing figure, let the radii of their pitch circles be represented in order by r,, r,, r,... r., beginning from the driving wheel; and let a, represent the perpendicular distance of the driving

He pressure from the centre of that

wheel, and a, that of the driven

pressure or resistance from the centre of the last wheel of the train; U, the work done upon the first wheel, u, the work yielded by the second wheel to the third, u, that yielded by the fourth to the fifth, &c., and U, the work yielded by the last or nth wheel upon the resistance, then is the relation between U, and u, determined by the modulus (equation 252), it being observed that the point of application of the resistance on the third wheel is its point of contact b with the third wheel, so that in this case a,=r,

These substitutions being made, and L, being taken to represent the distance between the point b and the projection of the point a upon the third wheel, we have

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To determine, in like manner, the relation between u, and u,, or the modulus of the third and fourth wheels, let it be observed that the work u, which drives the third wheel has been considered to be done upon it at its point of contact b with the fourth; so that in this case the distance between the point of contact of the driving and driven wheels and the foot of the perpendicular let fall upon the driving pressure from the centre of the driving wheel vanishes, and the term

See note p. 266.

which involves the value of L, representing that line disappears from the modulus, whilst the perpendicular upon the driving pressure from the centre of the driving wheel becomes 7. Let it also be observed, that the work of the fourth wheel is done at the point of contact c of the fifth and sixth wheels, so that the perpendicular upon the direction of that work from the axis of the driven wheel is r. We shal thus obtain for the modulus of the third and fourth wheels,

1

u = {1+◄ (+) sin. + La sin., },+N,S,.

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In which expression L, represents the distance between the point c and the projection of the point b upon the fifth wheel.

In like manner it may be shown, that the modulus of the fifth and sixth wheels, or the relation between u, and u, is

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and that of the seventh and eighth wheels, or the relation between u, and u,

u1+N..

= {1+(+) sin. p+ of sin. o. },+N.. S.;

and that, if the whole number of wheels be represented by 2p, or the number of pairs of wheels in the train by p, then is the modulus of the last pair,

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In which expressions the symbols N,, N,, N,... Np, are taken to represent, in respect to the successive pairs of wheels of the train, the values of that function (equation 247), which determines the friction due to the weights of those wheels; and each of the symbols L, L,, L,... Lp, the distance between the point of contact of a corresponding pair of wheels and the projection upon its plane of the point of contact of the next preceding pair in the train; whilst the symbols n,, n,, n,... np, represent the numbers of teeth in the wheels;,,,... p, the radii of their pitch circles; and S, S,, S,... Sp, the spaces described by their points of

contact a, b, c, &c. whilst the work U, is done upon the first wheel of the train.

Let us suppose the co-efficients of u,, u,, u,... U,, in these moduli to be represented by (1+μ,), (1+μ2), (1+μ,). . . . (1+); they will then become

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Eliminating u,, u,, u... up, between these equations, we shall obtain an equation of the form

U1=(1+,) (1+2) (1+μ,). . . (1+")U2+N.S.

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(277),

NS=N,S,+(1+μ‚,)N,S,+(1+μ‚)(1+μ ̧)N ̧S ̧+...

where

+(1+μ,)(1+~2).

....

(1+"p−1)Np Sp. . (278).

....

Now let it be observed, that the space described by the first wheel, at distance unity from its centre, whilst the space S, S, is.described by its circumference, is represented by and

S

that this same space is represented by if S represent the

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space described in the same time by the foot of the perpendicular a,, or the space through which the moving pressure may be conceived to work during that time; so Also let it be observed that the space de

that

S S

ra

scribed by the third wheel, at distance unity from its centre, is the same with that described at the same distance from its centre by the second wheel, so that

S, S,

1

=

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; in like

manner that the spaces described at distances unity from their centres by the fourth and fifth wheels are the same, so

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Multiplying the two first of these equations together, then the three first, the four first, &c., and transposing, we have

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Substituting these values of S,, S,, &c. in equation (278), and dividing by S, we have

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or if we observe that the quantities,,,,,, are composed of terms all of which are of one dimension in sin. o, sin. . sin.,, &c. and that the quantities N,, N,, N,, &c. (equation 247) are all likewise of one dimension in those exceedingly small quantities; and if we neglect terms above the first dimension in those quantities, then

N=((){N.+())N. + (7)N+

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If in like manner we neglect in equation (277) terms of more than one dimension in 1929 &c. we have

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