the rudder is adjusted. It usually points forward; that is, in the contrary direction to the rudder itself. In ships of war the tiller is usually put over, or moved to one side or to the other, by means of a pair of obliquely-acting twofold tackles, made of raw hide ropes, which haul it respectively to starboard (that is, towards the right) and to port (that is, towards the left), when required. The hauling parts of both tackles are guided by fixed pulleys so as to be wound in opposite directions round one barrel, which is turned by means of the steering-wheel.*

Fig. 156 A is a plan of this combination. A is the rudder-head; A B, the tiller, shown as amidships, or pointing right ahead; DB F G is the starboard tiller-rope; D' B F' G', the port tillerrope. These ropes are made fast to eye-bolts at D and D'; at B they are rove through blocks that are secured to the tiller; at F and F' they are led round fixed pulleys; and G and G'are their hauling parts, which are led, by means of pulleys which it is unnecessary to show in the figure, to the barrel of the steering-wheel.

A b is the position of the tiller when put over about 40° to starboard; and the corresponding positions of the tiller-ropes are Db F G and D' b F G'.

In order that the tiller-ropes may never become too slack, it is necessary that the sum of the lengths of their several parts should be nearly constant in all positions of the tiller; that is to say, that we should have, in all positions,

D b + b F + D ́ b + b F' nearly = 2 (D B + B F).

That object is attained, with a rough approximation sufficient for practical purposes, by adjusting the positions of the points D, D', and F, F, according to the following rule:

RULE.-About A, with the radius A B, describe a circle. Make

A C

=

2

3

A B; and through C, perpendicular to A B, draw a

straight line cutting that circle in D and D'.

These will be the points at which the standing parts of the ropes are to be made fast.

1

Then produce A B to E, making B E = -, A B; and through E,

12

perpendicular to A B E, draw FE F, making E FEF=

CD; F and F' will be the stations for the fixed blocks.

5

4

When the angle B A b is about 40°, the sum of the lengths of the parts of the ropes is a little greater than when the tiller is amidships; but the difference (which is about one-50th part of the length expressed in the preceding equation) is not so great as to

See Peake's Rudimentary Treatise on Shipbuilding, second volume, pp. 66, 162; also Watts, Rankine, Napier, and Barnes On Shipbuilding, p. 202.

cause any inconvenient increase of tightness. For angles not exceeding 30° the approximation to uniformity of tightness is extremely close.

SECTION VIII-Hydraulic Connection.

207. General Nature of the Combinations.-The kind of combinations to which the present section relates are those in which two cylinders fitted with moveable pistons are connected with each other by a passage, and the space between the pistons is entirely filled with a mass of fluid of invariable volume.

Any liquid mass may be treated, in most practical questions respecting the transmission of motion, as if its volume were invariable, because of the smallness of the change of volume produced in a liquid by any possible change of pressure. For example, in the case of water, the compression produced by an increase in the intensity of the pressure to the extent of one atmosphere (or 14.7 lbs. on the square inch), is only one-20,000th part of the whole volume. (See Article 88, page 75.)

The volume, then, of the mass of fluid enclosed in the space between two pistons being invariable, it follows that if one piston (the driver) moves inwards, sweeping through a given volume, the other piston (the follower) must move outwards, sweeping through an exactly equal volume; otherwise the volume of the space contained between the pistons would change; and this is the principle upon which the comparative motion in hydraulic connection depends.

208. Cylinders, Pistons, and Plungers.-A piston is a primary piece, sliding in a vessel called a cylinder. The motion of the piston is most commonly straight; and then the bearing surfaces of the piston and cylinder are actually cylindrical, in the mathematical sense of that word.

When the motion of a piston is circular, the bearing surfaces of the piston, and of the vessel in which it slides, are surfaces of revolution described about the axis of rotation of the piston; but that vessel, in common language, is still called a cylinder, although its figure may not be cylindrical.

A plunger is distinguished from an ordinary piston in the following way:The bearing surface of a cylinder for a plunger consists merely of a collar, of a depth sufficient to prevent the fluid from escaping; and the plunger slides through that collar, and has a bearing surface of a length equal to the depth of the collar added to the length of stroke; so that during the motion different parts of the surface of the plunger come successively into contact with the same surface of the collar. On the other hand, an ordinary piston has a bearing surface of a depth merely sufficient to prevent the fluid from escaping; and the cylinder has a bearing surface of a

length equal to the depth of that of the piston added to the length of stroke; so that during the motion the same surface of the piston comes into contact successively with different parts of the surface of the cylinder. For example, in fig. 157, A is a plunger, working through the collar B in the cylinder C; and in fig. 158, A is an

ordinary piston, working in the cylinder B. The action of plungers and of ordinary pistons in transmitting motion is exactly the same; and in stating the general principles of that action, the word piston is used to include plungers as well as ordinary pistons.

The volume swept by a piston in a given time is the product of two factors-transverse area and length. The transverse area is that of a plane bounded by the bearing surface of the piston and cylinder, and normal to the direction of motion of the piston, so that it cuts that surface everywhere at right angles. In a straight-sliding piston that plane is normal to the axis of the cylinder; in a piston moving circularly, it traverses the axis of rotation of the piston: in other words, the area is that of a projection of the piston on a plane normal to its direction of motion.

When the motion of the piston is straight, the length of the volume swept through is simply the distance moved by each point of the piston. When the motion is circular, that length is

the distance moved through by the centre of the area of the piston.*

So long as the transverse area and length of the space swept by a piston are the same, it is obvious that the form of the ends of that piston does not affect the volume of that space.

When the space in the cylinder which contains the fluid acted on by a piston is traversed by a piston-rod, the effective transverse area is equal to the tranverse area of the piston, with that of the rod subtracted. For example, in fig. 158, the upper division of the cylinder is traversed by the piston-rod C, working through the stuffing-box D; hence the effective transverse area in that division of the cylinder is the difference between the transverse areas of the piston A and rod C. In the lower division of the cylinder, where there is no rod, the whole transverse area of the piston is effective. A trunk acts in this respect like a piston-rod of large diameter.

209. Comparative Velocities of Pistons.-From the equality of the volumes swept through by a pair of pistons that are connected with each other by means of an intervening fluid mass of invariable volume, it obviously follows that the velocities of the pistons are inversely as their transverse areas.

The transverse areas are to be measured, as stated in the preceding Article, on planes normal to the directions of motion of the pistons; and when the motion of a piston is circular, the velocity referred to in the rule is that of the centre of its transverse area.

Let A and A' denote the transverse areas of the two pistons marked with those letters in fig. 159, page 224, and v and v' their

velocities; then their velocity-ratio is

v

=

A

v A"

As the velocity-ratio of a given pair of connected pistons is constant, the combination belongs to Willis's Class A.

210. Comparative Velocities of Fluid Particles.—It may sometimes be required to find the comparative mean velocities with which

To find the distance of the centre of a plane area from an axis in the plane of that area: divide the area, by lines parallel to that axis, into a number of narrow bands; let dx be the breadth of one of those bands, and y its length; then y d x is the area of that band; and y dx is the whole area. Let x be the distance from the axis to the centre of the band y dx; then x y d x is the geometrical moment of that band, and x y d x is the geometrical moment of the whole area relatively to the axis; which moment, being divided by the area, gives the required distance of the centro of the area from the axis, viz.,

the fluid particles flow through a given section of the passage which connects a pair of pistons; it being understood that the mean velocity of flow through a given section of the passage denotes the mean value of the component velocities, in a direction normal to that section, of all the particles that pass through it. From the fact that in a given time equal volumes of fluid flow through all sectional surfaces that extend completely across the passage, it follows that the mean velocity of flow through any such section is inversely as its area (a principle already stated in Article 88, page 76); and this principle applies to all possible sections, transverse and oblique, plane and curved.

For example, in fig. 159, let B denote the area of a transverse section, B B, of the passage which connects the two cylinders, and

u the mean velocity with which the particles of fluid flow through that section; then v, as before, being the velocity of the piston whose transverse area is A, we have

Also, let C denote the area of an oblique section, C C, of the passage, and w the mean component velocity of the fluid particles in a direction normal to that section; then

211. Use of Valves-Intermittent Hydraulic Connection.-Valves are used to regulate the communication of motion through a fluid