figure under an exceed ingly portable and convenient form. The axle or drum of the capstan is composed of two parts of different diameters. One extremity of the cord is coiled upon one

of these, and the other, in an oposite direction, upon the other; so that when the axle is turned, and the cord is wound upon one of these two parts of the drum, it is, at the same time, wound off the other, and the intervening cord is shortened or lengthened, at each revolution, by as much as the circumference of the one cylinder exceeds that of the other. In thus passing from one part of the drum to the other, the cord is made to pass round a moveable pulley which sustains the pressure to be overcome.

To determine the modulus of this machine, let u, and u, represent the work done upon the two parts of the cord respectively, whilst the work U, is done at the moving point of the machine, and U, yielded at its working point.

Then, since in this case we have a body moveable about a cylindrical axis, and acted upon by three pressures, two of which are parallel and constant, viz. the tensions of the two parts of the cord; and the point of application of the third is made to revolve about the axis, remaining always at the same perpendicular distance from it; it follows (by equation 198), that, for n revolutions of the axis,

a, and a, representing the radii of the two parts of the drum, a, the constant distance at which the power is applied, and the radius of the axis.

* A figure of the capstan with a double axle was seen by Dr. O. Gregory among some Chinese drawings more than a century old. It appears to have been invented under the particular form shown in the above figure by Mr. G. Eckhardt and by Mr. M'Lean of Philadelphia. (See Professor Robinson's Mech. Phil. vol. ii. p. 255.)

Also, since the two parts of the cord pass over a pulley,

and the pulley is made to revolve under the tensions of the two parts of the cord, u, being the work of that tension which preponderates, we have (by equation 181), if S represents the length of cord which passes over the pulley,

a representing the radius of the pulley, P, the radius of its axis, W its weight, and the inclination of the direction of the tensions of the two parts of the cord to the vertical, the axis of the pulley being supposed horizontal, and the two parts of the cord parallel. Now t,=

Sub

ՂԱ t= 2ηπα, 2ηπα, stituting these values, and multiplying by 2nra,, we have

Изаг
a,

(200).

Since the tensions t, and t, of the two parts of the cord, and the pressure P, overcome by the machine, are pressures applied to the pulley and in equilibrium, and that the points of application of t, and P, are made to move in directions opposite to those in which those pressures act, whilst the point of application of t, is made to move in the same direction; therefore (Art. 59.),

Eliminating u, and u, between this equation and equation (200), we have

Substituting these values in equation (199), and reducing,

Substituting their values for A, A,, B, B,, neglecting terms involving more than one dimension of

sin. E

a1 a,

reducing, we obtain for the MODULUS of the machine,

&c. and

From which expression it is apparent that when the radii a, and a, of the double axle are nearly equal, a great sacrifice of power is made, in the use of this machine, by reason of the rigidity of the cord.

THE HORSE CAPSTAN, OR THE WHIM GIN.

185. The whim is a form of the capstan, used in the first operations of mining, for raising materials from the shaft and levels by the power of horses, before the quantity excavated is so great as to require the application of steam power, or before the valuable produce of the mine is sufficient to give a return upon the expenditure of capital necessary to the erection of a steam engine. The construction of this machine will be sufficiently understood from the accompanying figure. It will Le observed that there are two ropes wound upon the drum in opposite directions, and which traverse the space

between the capstan and the mouth of the shaft. One of these carries at its extremity the descending (empty) bucket, and is continually in the act

of winding off the drum of the capstan as it revolves; whilst the other, from whose extremity is suspended the ascending (loaded) bucket, continually winds on the drum. The pressure exerted by the horses is that necessary to overcome the friction of the different bearings, and the other prejudicial resistances, and to balance the difference between the weight of the ascending load, bucket, and rope, and that of the descending bucket and The rope, rope. in passing from the capstan to the shaft, traverses (sometimes for a considerable distance) a series of sheaves or pulleys, such as those shown in the accompanying figure.

Let now a represent the radius of the drum on which the rope is made to wind, and n the number of revolutions which it must make to wind up the whole cord; also let represent the weight of each foot of cord, and the angle which the capstan has described between the time when the ascending bucket has attained any given position in the shaft and that when it left the bottom; then does a, represent the length of the ascending rope wound on the drum, and therefore of the descending rope wound off it. Also, let W represent the whole weight of the rope; then does W-a represent the weight of the ascending rope, and pa, that of the descending rope, both of which hang suspended in the shaft. . Let P, represent the load raised at each lift in the bucket, and w the weight of the bucket; then is the tension upon the ascending rope at the mouth of the shaft represented by W-a,+P2+w, and that upon the descending rope by pa,+w.

Let, moreover, P. and p, represent the tensions upon these ropes after they have passed from the mouth of the shaft, over the intervening pulleys, to the circumference of the capstan.

Now, since the tension upon the ascending rope, which is W−μa2+P2+w at the mouth of the shaft, is increased to P, at the capstan, and that the tension upon the descending rope, which is p, at the capstan, is increased to paw at the mouth of the shaft, if we represent by (1+A) and B the constants which enter into equation 180 (Art. 174.), we have,

by that equation (observing that U1=P,S, and U,=P,S,, sc that S, disappears from both sides of it),

and

p ̧=(1+A)(W+P2+w—μɑ‚§)+B, . . . (202),

μa,+w=(1+A)p,+B.... (203).

Multiplying the former of the above equations by 1+A, adding them, transposing, dividing by (1+A), and neglect ing terms of more than one dimension in A and B,

P1—P1=(1+A)(W+P2)+2Aw+2B−2μɑ„§.

Now U, in equation (193) represents the work of the resultant of p, and p, during n revolutions of the capstan, it therefore equals the difference between the work of P1 and that of p, (see p. 198).

2nπ

.. Ur=a,/ {(1+A)(W+P2)+2Aw+2B—2μa,0} do=

0

{(1+A)(W+P2)+2Aw+2B}(2n≈a,)—μ(2nĩa,)3 ; ..U,=(1+A)U,+ {(1+A) W+2Aw+2B—μS,}S,... (204) ; observing that 2nлa,=S,, and that P,S,=U,.

P

Now, let it be observed that the pressures applied to the capstan are three in number; two of them, p, and p, being parallel and acting at equal distances a, from its axis; and the third, P., being made to revolve at the constant distance a, from the axis (P, representing the pressure of the horses, or the resultant of the pressures of the horses, if there be more than one, and a, the distance at which it is applied); so that equation 193 (Art. 182.) obtains in respect to the pressures P, P, P.; a, being assumed equal to a

Substituting p, and p, for P, and P, in equation (194), RL=aa,(p2+p2)—a‚(p‚—p‚) (a cos. 0—ß sin. 0);

..

RLdo

ƒ =

RLd0=aa, (p2+P2) do — a2 (Ps—P2)

(a cos. 0-3 sin. 0) do.