F WE watch a complicated machine in operation, we observe a seemingly great variety of motions. Some of the parts are rotating on axles or shafts; others are sliding in guides; certain connecting links have a curious complicated movement, which appears to be partly a sliding and partly a rotating, or turning; still other links have, perhaps, a screw-like motion. These motions of machine parts, and of bodies in general, furnish an interesting and useful subject for analysis and study.

All motions of rigid bodies may be primarily divided into two classes: plane motion and non-plane motion. A body has plane motion when it moves in such a manner that each point remains always in one plane during the motion, the planes in which the different points of the body move being parallel. Take, for example, the flywheel or pulley; it is evident that any point of the wheel always remains in a plane which is perpendicular to the shaft, and that the planes of the different points are parallel. A body having non-plane motion moves in such a manner that the path of any point is a curve in space; that is, a curve which does not lie in a plane. A familiar example of non-plane motion is the motion of a nut on a bolt; as the nut advances, it turns on the screw so that every point of it is describing a helical path. The universal joint is another instance of nonplane motion. By far the greater part of the motions met with in practice are plane

motions, and in this article we shall restrict our study to motions of that class; so that, whenever we speak of motion, or say a body moves, it will be understood that plane motion is meant.

Every point of a moving body describes a curve (or a straight line) in the plane of its motion; this curve we shall call the pointpath. The direction of a point's motion at any instant is given by the tangent to the point-path. Thus, suppose the point a of the body A, Fig. 1, moves in the curved path aa'. When the point is at a, the line ab-tangent to

the path at a

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shows the direction in which it is moving just at that instant. When the point is at a,, the tangent a, b, shows its direction of motion; likewise, the tangent a, b, shows the direction when the point is at a2. It must be carefully noted that the tangent gives the direction of motion just at the particular instant that the point occupies the given position. If the point moves ever so little from its position at a, the tangent changes its direction.

Since, in plane motion, all points are moving in parallel planes, the motion of the body, whatever its size or shape, will be determined by the motion of any thin slice parallel to the plane of motion. To illus

trate this point, suppose we move a book in any manner upon the surface of a table; then the motion of the book as a whole will be represented perfectly by the motion of the thin cover next the table. This method of representing a solid object by a thin section is of great service, for we may consider the rotation of the body about an axis replaced by the rotation of the plane section about the point where the axis pierces it.

In order to obtain a knowledge of the motion of a body, we need to know all

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about the motions of two of its points. In Fig. 2, A represents a thin slice of a body. Suppose we stick an axis or pin p through the body, piercing the slice at o; the only motion the body can have is a rotation about

the pin as a center, and therefore any point of the slice, as a, must describe a circular arc about o as a center. The direction of the motion of the point a, when it is in the position shown, is along ab, the tangent to the arc, and the direction of motion of any other point m is along the tangent m n to its circular path. It is a geometric property of the circle that the radius oa which joins the center o to the point a is perpendicular to the tangent ab; likewise, om is perpendicular to the tangent mn. These considerations lead to the following important principle: The line joining the center of rotation to a given point is perpendicular to the direction of the point's motion; conversely, the center of rotation lies in a line drawn through the moving point perpendicular to the direction of the point's motion; that is, perpendicular to the point's path.

This principle enables us to determine the motion of any rigid body, provided we know the motions of two of its points. Suppose B, Fig. 3, to be a moving body, and that we know that the two points a and m are moving in the directions shown by the lines ab and mn, which are, respectively, the tangents to the paths of a and m. If, now, we draw through a the line ac perpendicular to ab, this line must pass through the center about which the body is rotating. Likewise, the line mp perpendicular to mn must also pass through the center, which must, therefore, lie at the intersection o of

the two lines. It must not be supposed that the body continues to rotate about the center o for any length of time; this can only happen when the points a and m move in circles with o as a center, which is not necessarily the case. In the most general case, the points a and m may move in any curves whatever. Let us assume in the present instance that a is moving in the path ef and that m is moving in the path gh. When a arrives at a, it is moving in the direction a,b,; at the same instant the point m is at m, and is moving in the direction m1n. Drawing perpendiculars to a, b1 and m11, through a, and m1, respectively, we find the center of rotation, at this particular instant, to be at 01.

A special case of frequent occurrence is that in which the paths of the two given points a and m are parallel straight lines, as shown in Fig. 4. The perpendiculars ac and mp are also parallel; that is, their intersection lies at an infinite distance from the body. A motion of this character is called a translation.

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We see that there are three (and only three) kinds of plane motion. They are : (1) Rotation about a fixed or permanent axis, or center, as shown in Fig. 2. (2) Rotation about a center, or axis, which is constantly changing its position, Fig. 3. (3) Translation, or rotation, about a center at an infinite distance, Fig. 4. An example of each of these motions exhibited in the familiar mechanism of the steam-engine. The crank and fly-wheel rotate about a fixed axis, the center line of the shaft. The connecting rod rotates about a center which is itself moving, and the cross-head has a motion of translation.

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FIG. 3.

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So far we have considered such points as o, Fig. 2, and o, Fig. 3, merely as centers about which their respective bodies are rotating; there is, however, another conception of these points which is important. In Fig. 2 we have assumed that the body 4 is moving and that the body B is at rest. Now, motion is only relative, and, so far as these two bodies are concerned, it makes no difference in their relative motion which of the two is moving and which is at rest. For example, we can make point a of A coincide with point e of B, either by rotating A in the direction of the hands of a watch through the angle a o e, or by rotating B through the same angle in the bopposite direction,

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FIG. 4.

in the meantime keeping A stationary. Similarly, in Fig. 3, the body B is moving relatively to some other body, which is assumed to be stationary, and the instantaneous rotation of B about the center o may be replaced by an equal rotation of the second body about o, in the opposite direction. In both cases the point o is the center of rotation, whichever of the bodies is considered as moving; that is, the center o belongs to both of the bodies. Suppose that the bodies A and B, Fig. 2, are moving relatively to each other, and that both are moving relatively to a third body, i. e., on the surface of a table or on a floor. The point is common to A and B, and has the same motion whether we consider it as belonging to A or to B. The same is true if is an instantaneous center; for the instant under consideration, it is the point (and the only point) which is common to A and B, and which has the same motion whether considered as a point of A or a point of B.

Let us now apply the principles so far developed to a concrete mechanism. In Fig. 5 is shown a combination of four bodies, a, b, c, and d ; modifications of this mechanism are frequently used in machinery. We will consider the link d as stationary, and study the motions of the other three links relative to it. Links a and c being joined directly to d, must rotate about the joints (a d) and (cd), respectively, which are therefore permanent centers. Considering links a and b, if one moves relatively to the other, it must be about the joint (ab); for a like reason, any motion of b relative to c, must be

a rotation about (be). Thus, the four centers of the adjacent links are the four joints, and are permanent centers. To find the center of the motion of b relative to d, we observe that there are two points of b, viz., (ab) and (bc), which we know are moving at right angles to the links a and c, respectively. The point (ab) belongs both to link a and to link b. Considered as a point of a, its path must be a circle with (ad) as a center, and its direction of motion must be perpendicular to a. Considering (ab) now as a point of b, the center of rotation of b must lie in a line perpendicular to the direction of motion of a b, which line must be a continuation of the link a. Likewise, the center must lie in a line perpendicular to the direction of motion of (bc), which is a continuation of link c. Hence, the center (bd) of the motion of b, relative to d, lies at the intersection of the continuations of links a and c. This center (bd) may be looked at from two points of view. If we suppose the links b and d enlarged so as to include (bd)—for this purpose we may imagine two sheets of paper, one pasted on b and the other on d— the point (bd) is the center about which b is rotating, considering d stationary; conversely, it is the center about which d is for the instant rotating, considering b stationary. If, however, both b and d are moving, the link a, for example, being stationary,

then the point (bd) is the point common to links b and d, which has the same motion whether considered as a point of b or as a point of d. The instantaneous center (a c) of the motion of a, relative to c, is found at the intersection of the continuations of links b and d; it is the point common to the links a and c.

An inspection of Fig. 5 shows that the three centers belonging to any three links lie in a straight line; thus the centers (a c), (a d), and (c d) of the links a, c, and d, lie

in the line (a c)-(c d). It can easily be proved that this must be true of any three bodies having relative motion. Thus, in Fig 5, if d is considered as stationary, every point of a must rotate about (a d) as a center, and every point of c must rotate about (c d) as a center. Now (a c), as we have seen, is a point common to a and c. Considered as a point of a, it is rotating about a d, and, therefore, the perpendicular to its direction of motion must pass through (a d); considered as a point of (c), (a c) is

rotating about (cd) as a center, and the perpendicular to the direction of its motion must pass through (cd). Since there can be but one perpendicular to a line at a given point, and since this perpendicular passes through both (ad) and (cd), it must be the line joining them. Therefore, (a c) lies on the line adjoining (a d) and (cd).

The use of the instantaneous center in determining the velocities of the moving links of a mechanism will be considered in a future article.

HE demand nowadays is for increased speed in all modes of transit. The telegraph and the telephone have within the last few years brought the various cities and towns nearer and nearer together, as regards communication. This has satisfied business men to a great extent, facilitating immensely, as it has done, the transaction of business. A prospective buyer doesn't have to wait for an agent or drummer to come along; he can talk directly with the firm itself in a more desirable and expeditious manner than by letter, the expense of such a proceeding being curtailed in many cases by the use of special telegraph codes.

But the business man is not content with even this. He wants to be able to go rapidly from place to place in person, and this desire is shared by most other people. Very few persons take a railroad journey for pleasure, but simply because they want to get to some other place. In case this may be regarded as a truism, we would instance a converse case-sea-travel, which many indulge in for the sake of the journey itself.

Railroad speeds have, on some roads, increased greatly of late years, but not so much as many people might imagine. Heavier trains are hauled nowadays, but the speeds have not shown a very marked increase, especially in the matter of schedule time. This is, on second thought, not a matter for surprise. First, because the railroads are more crowded, especially in

Europe, and secondly, because the engine of to-day is very similar to what it was 30 years ago. It has been enlarged all around, 'tis true; larger cylinders are being used, larger boilers, higher steam pressures, and all parts are made stronger and heavier, but in general design it remains very much the same. The engines of to-day of course take heavier loads, and work with greater economy; the average speeds, too, are perhaps somewhat higher in the majority of cases, although this is greatly due to better signalling—the adoption of the block system and the interlocking apparatus; the use of automatic continuous brakes, too, permit of these higher speeds being employed. But the maximum speeds of the engines themselves 25 years ago, compare very favorably with those of to-day, except in one or two exceptional cases.

In hinting at retrogression, the writer has in mind a certain English road whose mainstay is its passenger traffic. This road now runs the majority of its through expresses from London to a certain town in 70 minutes-one a day, each way, however, being timed to do it in 65 minutes (we italicize the "timed" advisedly). Now, 30 years ago the schedule time was 60 minutes, and they made it, too. The traffic was less crowded, of course, and the cars were smaller, but the loads were proportional to the power and weight of the engines. Again, during the " race to the North" that

has been indulged in during the last few summers in Great Britain, the majority of engines employed have been old types. Some noteworthy running on the L. N. W. was done by Webbs' 4-coupled Precedent class with 78-inch drivers. These engines, however, are practically the same as they were 20 years ago. As time progressed, and the rival concern began to make the pace hot, the above company trotted out the old Lady of the Lake class, first built by Ramsbottom as much as 35 years ago. They had a single pair of drivers, 911⁄2 inches diameter, 16- by 24-inch cylinders, 1,100 square feet of heating surface, and only weighed 60,500 pounds, 25,750 of which was on the drivers. The boiler was 48 inches, outside diameter, and its center was but 6 feet 6 inches above raillevel. With a load of not more than about 6 of their 6-wheeled coaches, these engines can hold their own to this day. The East Coast route relied chiefly on a type of engine first built nearly 30 years ago, and practically the same at the time we are speaking of. These also had single drivers, 97 inches diameter, however, in this case.

It is often stated in print that the G. W. R. (England) in 1840 ran trains at 50 miles an hour, and had engines that made 80 miles per hour without a train. The former statement is true enough, but the latter must not be taken too literally. Few men would like to run a "light" engine (that is, one without a train) at top speed. It is necessary to have a car or two behind to steady her. Doubtless the speed mentioned was attained with a very light load on a down grade.

There are not many regular trains running nowadays at a schedule speed of more than 53 miles per hour. Why shouldn't they be timed at 60? By far the fastest train in the world is the one that runs from Camden to Atlantic City in the summer months. This makes the 55 miles, start to stop, in from 47 to 50 minutes, regularly. Still, the maximum speeds are not always attained by expresses. Often the fastest running is done by the stopping trains. The highest degree of skill and judgment is by no means called forth solely on the fastest and most important trains, where there is a through run, a clear road, and not much likelihood of overloading. Everything is done, in fact, to ensure their getting through "on time." But with a stopping train much time is often lost at stations by tardy baggagemen, or through late connections, or in waiting for the mails. If, under these circumstances,

the engineman doesn't put his best leg foremost and make up a lot of the lost time, he doesn't advance in the good graces of his superior officers. On such trains one is likely to experience much faster bursts of speed than on the more prominent and popularly-known expresses; in fact, on the stopping trains the running often becomes right-down reckless.

As regards the accelerating of the journey as a whole, there are many points to be considered, militating for or against it, which we propose to consider briefly and in a general way, classifying them as follows:

1. The nature of the road; its freedom from curves and grades.

2. The quality of the track itself. 3. The condition of the rolling stock. 4. The operation and management of the traffic department.

5. The wind and weather.

6. Safety appliances. (The block system of signaling. Interlocking apparatus. Continuous brakes. Double tracks.)

7. The capacity of the engine itself.

1. The nature of the work: Great skill is now evinced in railroad location, avoiding sharp curves and grades as far as possible. They couldn't be expected to know it all in those early days; we've all had to learn, and we have also profited by other people's mistakes. There is little doubt that most of the earlier lines, if laid out afresh, would be easier and shorter, a freer use being made of cuts, tunnels, and embankments.

A great deal depends upon the nature of the country passed through, and also on the financial standing of the company. Sometimes the choice lies between sharp curves and grades on the one hand, and cuts, tunnels, and embankments on the other. If, however, the traffic is to be predominatingly passenger, and money is plentiful, we may assume that the road will be pretty easy-that is, comparatively free from curves and grades and then a certain acceleration of schedule time may be counted upon.

2. The quality of the track itself will prove an important feature in the raising of train speeds. We want the engine and cars to move forward all the time; all motion in any other direction is wasted. It seems obvious that all lateral swaying, vertical pitching, etc. will detract to some extent from the speed.

A great deal of unsteady running is due to the track itself. It may be ballasted badly, either an unsuitable material being used or