EX. 2.-Describe a circle of 3" diameter, and mark a point, P, outside the circumference 1" away, and a point, Q, inside the circumference" away. Then draw curves equidistant from point P and the circle, and point Q and the circle.

EX. 3.-Draw two circles of 12" and " radius, centres 3" apart, and a line parallel to the line joining the centres of the circles 21" away from the centre of the larger circle. Then find the centre of the circle touching the given circles and the given line.

EX. 4.-Draw any three circles of unequal diameters not touching or cutting, and find the centre of the circle touching all three circles externally.

EX. 5.-Draw any three circles of unequal diameters not touching or cutting, and find the centre of the circle touching all three circles and including them.

PROBLEM XXXIII. (Fig. 39).-A pendulum of a given length swings uniformly through a given angle. A point uniformly descends the pendulum from the top to the bottom during one complete swing. Trace the locus of the point.

Let A B and A C represent the pendulum at beginning and end of the swing. Then the travel of the point from A towards B is the length A B, and of the pendulum B bob the arc, BC, both uniformly and in the same time. Divide both travels into the same number of

6

equal parts, say eight, and mark as shown. Draw the pendulum in the different positions, A l', A 2′, . . . A 7'. Then when the pendulum has reached the position A 3′ the point will have travelled the distance A 3, therefore, with centre A and radius A 3 draw an arc to meet the line A 3' in the point, P, which will be one point in the locus of the point. Proceed in the same way for other points and draw the curve as shown.

PROBLEM XXXIV. (Fig. 40).-To draw the path of a point in a link, one end of which moves in a circle while the other end moves in a straight line.

(This is the combination of crank, connecting-rod, and guide bars, so common in steam engines.)

Let A B be the link, the end B moving in the circle of centre C, while the end A (the piston end) moves in the straight line through A C.

Divide the circle into any number of equal parts, and mark 1, 2, . . . 12, as shown. (Twelve is the most convenient number, as a quadrant can be divided into three by marking off the radius from each end.) With the length, A B, of the link as radius, and the points 1, 2, . . . 12 as centres, cut the line A C in the points 1', 2′, 3′, . . . 12′, and join the points 2′, 2; 3′, 3, . . thus drawing the link in each of the twelve positions. Then measure off from either end of the link in each position the distance from that end of the point, the locus of which is

B

...

required, thus obtaining the points P2, P3,... through which the curve of the complete path is drawn.

It is interesting to note how the path of different points in the rod changes from a straight line at the guides (the end A), through oval curves of different degrees of convexity, until it reaches a circle at the crank end, B. Points in an extension of the rod beyond B, travel in oval paths, the long axes of which are at right angles to the line A C. Notice also that the piston end, A, does not move uniformly with an uniform movement of the crank.

PROBLEM XXXV. (Fig. 41).-To trace the locus of a point in the linkwork known as Watt's Simple Parallel Motion.

This linkwork consists of two links, A B and CD, pivoted at A and D, and having their other ends connected to a shorter link, BC, to a point P in which the piston-rod is attached. The link A B is the engine beam, and the link CD the radius bar. When the links are equal, the point P is in the middle of the short link, BC. This linkwork is also used on Richard's Engine Indicator

With centres A and D, and radii A B and DC respectively, draw arcs EBF and GCH. The end B must always move in the arc EBF and the end C in the arc G CH.

In problems of this kind it is always best to start by finding the limiting positions of the links. Suppose the link AB is moving upwards, then its limiting position is AE, when DC and C B are in one straight line. Find the point E by taking the sum of the lengths DC and CB as radius, from centre D. If the line AB now moves down, DC will continue to move upwards until the end C reaches the position G, where A B and

In a similar way the limiting

BC are in one straight line. bottom positions F and H are found. To find the path of P for the complete movement of the linkwork, draw the links in a convenient number of different positions, and mark the position of P. For example, if link A B moves to A1, then, with length of link BC as radius, and point 1 as centre, cut the locus of C (the arc HCG) in 1', and join the points 1 and l', then mark the point P as P. Notice that it is unnecessary to draw the links A B and CD, and that it is better to find a greater number of points at places where the curve changes in direction.

A very convenient method of obtaining points in the travel of the point P is to mark off the points BPC along the straight edge of a slip of paper, and the correct distances apart, and then if the paper be moved so that the point B is always on the circle EB F, and the point C on the circle G CH, the different posi

tions of the point P can easily be marked. Care must be taken to move the points B and C in the right direction, since they do not both always move in the same direction. This mechanical method may be very accurate, and admits of useful extension to similar problems.

In the application of this linkwork to steam engines, the travel of the point, P, does not exceed that part which approximates to a straight line.

EXAMPLES.

EX. 6.-A pendulum, 4 feet long, is moved from rest, and makes one-half swing to the left and one complete swing (through 40°) to the right, while a fly travels from the top to the bottom. If the travels are uniform trace the path of the fly. (Scale 1′′ 1 foot.)

=

EX. 7.-(a) A connecting-rod is 3' 6" long, the crank being 6" long. Trace the paths of points, 1' 3", from each end of the rod during one complete revolution of the crank. (b) Mark on the line of travel of the piston end of the connecting-rod, the distances representing the travel of that end, while the crank pin end travels uniformly. (c) Work the same problem when the connecting-rod is 1' 6" long, and notice how the motion of the piston is affected by the length of the connecting-rod. (Scale 1" 1 foot.)

=

EX. 8.—A parallel motion consists of two arms, 4′′ long, pivoted at their outer ends, and connected by a link 2′′ long. In the central position the arms are parallel, and the link is inclined to them at 60°. Draw the complete path of (a) the central point of the link, (b) a point" from one end. (Vict. U. Hon., 1891.)

=

=

= 2".

EX. 9.-Draw two lines, A P, CP, 23′′ and 11′′ long respectively, meeting at a point, P, so that angle APB 60°. Produce AP to B, and CP to D, so that AB 4", and CD: If A B and C D are links pivoted at A and C respectively, and P is a saddle which can travel along CD at two-fifths the speed it can move along AB, trace the locus of P. What is this curve? (S. & A. A., 1888.)

=

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EX. 10.-Draw a rhombus ABCD, sides 18" long, acute angles at B and D 45°, and mark a point, P, in BC 1" from C. Draw a circle of 118" diameter passing through C, such that the centre is on AC produced and beyond it. If the rhombus is a linkwork pivoted at A, trace the locus of the point P, when the joint, C, moves in the circumference of the circle. (S. & A. A., 1887.)

EX. 11.-A point O is 13" from the centre of a circle of 3"

radius. Determine the locus of the centres of circles bisecting the circumference of this circle and passing through the point O. (S. & A. A., 1891.)

(If the circles bisect the circumference of the given circle, they must pass through the ends of diameters. Therefore, by drawing a diameter in different positions, the centres of circles can be found, which would pass through the ends and the given point.)

Pantograph (Fig. 42).-This consists of an arangement of links in which two

points move in similar paths, and is thus capable of application for reducing and enlarging any given pattern.

Fig. 42.

B

A C is a link pivoted at A, the links B E, E D, and D C form a parallelogram with the part B C. If a line be drawn from A through any point, P, in B E and produced to meet a point, P', in CD or CD produced, then the points P and P' trace out similar paths, the locus of P' being the larger, for all positions of the linkage.

=

EX. 12.-Draw a pantograph as in Fig. 42 as follows:AC 4", BC= 17", BE=", BP", angle ABE = 60°, and trace the locus of P' when P moves in an equilateral triangle of 1" side.

E

Watt's Double Parallel Motion (Fig. 43).—This consists of a simple parallel motion (Fig. 41) added to a pantograph (Fig. 42). The simple links A B, BC, and CD cause the point P

Fig. 43.

to move for a short distance in a straight line, while the pantograph links, A B E, B C, C P', and P' E, give a similar movement to the point P'. The link AE is half the engine beam, the piston-rod being attached to P', and the pump-rod to P.

EX. 13.-Draw the linkwork of Watt's double parallel motion as in Fig. 43 as follows:-A E = 31′′, A B = 21′′, BC = 3′′, angle ABC 60°. Join the points A P' to give P, then find length of A B CP

=

CD knowing that

for all positions of the links.

Trace out the paths of P and P'

Scott-Russell's Parallel Motion (Fig. 44).-This is the