able aid is rendered to the correct comprehension of them by means of graphic representation. Graphic Representation of Magnitudes. — When we have two magnitudes bearing certain relations to each other, we can represent these relations by means of a plane curve; values of one magnitude being marked off along an axis, and ordinates set up at the points so obtained, to represent the corresponding values of the other. A curve drawn through the tops of the ordinates exhibits to the eye the relationship between the magnitudes, and is often of great use in the solution of problems. In the example previously considered the two magnitudes are Time and Velocity. LA Fig. 24. Take then two axes OX and OY. Along OX we will set off time. Now it is quite immaterial at what point of OX we begin. Suppose then we let O, or time zero, represent sec. before the body was let go. Mark off then Oa to represent on some scale ↓ sec., Oa is in the figure inch, the scale being 1 inch to I second. At a the velocity is zero, so a is a point on the curve, the ordinate being o. Now set off ac = 1 second, ac will be 1 inch long. I inch, so the velocity scale is 1 inch to 32 f.s. a second point on the velocity curve. e is then This we We said that the velocity varies uniformly. express graphically by drawing the velocity curve as a straight line from e to a. For then the height of the curve from OX varies uniformly. We can now by means of this curve determine the velocity at any instant during the second of motion, or we can determine the mean velocity. To determine the mean velocity or the mean value of any magnitude, when we have in the manner shown represented it by a curve, we have only to determine the mean height of the curve. In the present case the mean height is plainly bd, where b is midway between a and c, and .. Mean velocity=bd, and bd being inch, and the scale 32 f.s. to 1 inch, Mean velocity = 16 f.s. We must be careful to notice that b represents midtime of falling, not mid-height; so 16 f.s. is the velocity at the end of a half second. Units of Velocity.-We have so far reckoned velocity in feet per second, because the foot and second are the most usual units of space and time. But any units whatever of space and of time may be used, depending on which are most convenient in any particular case. It occurs rarely that a smaller unit than feet per second is necessary, but if so inches per second or feet per minute may be used. The latter of these is much used for reckoning piston speeds. In the motions of trains and ships it is customary to use larger units, and generally miles per hour is the unit selected. For trains, and bodies on land generally, the mile is the ordinary mile of 5280 feet. Hence, if we wish to interchange the units in expressing a velocity we use the following relation : I mile per hour = 5280 ft. per hour, = ft. per minute, which is generally the other unit. If we require the velocity in f.s., then I mile per hour=3=1.46 f.s. In the case of ships there are two distinct miles used, viz.— The ordinary mile as above, or The nautical mile of 6080 ft. Thus when the speed of a ship is given in miles per hour, it should be always stated which kind of mile is meant, since the difference is considerable. before we have I nautical mile per hour=98 Working as 101 ft. per min.; usually 101 is taken as sufficiently approximate. The expression "nautical mile per hour" is never used in practice, but is abbreviated to knot. So that in the preceding we should write I knot == etc. It is common to find "knot " used as if it represented "nautical mile," so we have speeds given as 14, 15, etc. "knots per hour." This is erroneous, and should be guarded against, not that in the present case any grave error is caused, but because the student, by never using a term in any other than its strict meaning, will save himself from falling into numerous difficulties. One effect of the difference in length of the nautical and land mile is to create a false impression regarding speeds of ships. Thus if a ship have a speed of 20 knots, then Turning. We consider next another simple kind of relative motion, viz. That in which two bodies are so connected that one can only move by turning round a centre fixed in the other and vice versa. The motion we call Turning, and the two bodies form a Turning Pair. The simplest example of a turning pair is a round rod fitting in a hole (Fig. 25). The rod has collars Fig. 25. which prevent endwise motion, and thus turning is the only relative motion possible. As practical examples we may take a propeller or crank shaft (Fig. 26) in its Fig. 26. Fig. 27. bearing, an eccentric and strap (Fig. 27), or a wheel on its axle. It appears in the case of a propeller shaft end wise motion is possible. But this is prevented by the thrust bearing at another part of the shaft. In crank shafts the journals, or parts of the shaft in the bearings, are sometimes turned smaller than the rest of the shaft, as Fig. 28, but it is best in marine engines to leave this also to the thrust block, such journals being nearly always a source of trouble. In turning, as in sliding, we often meet cases in which the motion is not strictly defined by the connection of the pieces. For example, a heavy shaft with the caps off the bearings would still revolve in them, being kept in place by its weight, and they form a turning pair (compare page 20). The only conditions the bodies must satisfy are that a circular projection on one must fit a hole in the other; and motion other than turning must in some, it matters not what, way be prevented. Fig. 28. In estimating the motion of a sliding piece we were at liberty to select any point on the piece, the motion of all points being identical. But this is no longer the case in a turning pair. For example, let C be a point in the moving body, the paper representing the fixed one; O is the centre of motion, i.e. the centre of the pin on the moving body and the hole in the fixed one, or vice versa. D C B Then during the motion C moves say to Fig. 29. C' in the circular arc CC'. Now the motion of C is not sliding, since CC' is not a straight line. But, taking for simplicity the case of uniform motion, we can take the curve as made up of a large number of small straight pieces, along each of which C slides in turn at a constant velocity, continually changing the direction of the velocity but not its magnitude. Such motion, although not sliding, can yet be measured in the same units as sliding; and we say the Linear Velocity of C is given by dividing the length of the arc CC' by the time occupied in describing it. C then moves at say v f.s., the instantaneous direction of its motion at any point of the arc being along the small piece of arc at that point, i.e. along the tangent. But now this velocity of the point C does not give us the velocity of the body. Because, for example, the |