draw DE perpendicular to A C, and C E parallel to BX; the intersection E will be a point in the parabola, and ED a tangent. Then parallel to CA, draw E F; this will be a normal, and a position of the ball-rod. From F, parallel to D E, draw F G, cutting CE produced in G; and from G, parallel to B Y, draw G, cutting EF produced in H; this will be a point in the evolute. To express this algebraically, let B C=y and CE=z be the co-ordinates of the parabola; and let B M = 2 and M H y' be those of its evolute. Then we have

=

= 3 z;

II. Another method of guiding the balls is to support them by means of a pair of properly curved arms, on which they slide or roll. On the top of the balls there rests a horizontal plate or bar, which communicates their vertical movements to the regulator.

III. Approximate Parabolic Governor.-In Farcot's governor, the rod E H, in its middle position, is hung from a joint, H, at the end of an arm, MH; this gives approximate isochronism. The co-ordinates of the point H are found by the rules already given. 362A. Loaded Parabolic Governor.-When the balls of a parabolic governor are guided in the second manner described in the preceding article, and support above them a plate or bar, to which their vertical movements are communicated, an additional load may be applied to them by means of that plate. Let A be the collective weight of the balls; B, the additional load; then the altitude corresponding to a given speed is greater than in the unloaded governor, in the ratio of A+ B: A; and the speed corresponding to a given altitude is greater, in the ratio of √(A + B): √ A; and by varying the load, the speed of the governor may be varied at will.

363. Isochronous Gravity-Governor (Rankine's).—In this form of governor (see fig. 259) the four centrifugal balls marked B are balanced, as regards gravity, about the joint A, on the spindle A M. D, D are sliders on the ball-rods; D C, D C, levers jointed to the sliders, and centred on a point in the spindle at C, and of a length D C = CA; G G, a loaded circular platform hung from the levers CD, CD, by links EF, EF; H, an easy-fitting collar, jointed to the steelyard lever H K, whose fulcrum is at K; L, a weight adjustable on this lever. This governor is truly isochronous; the altitude h of a revolving pendulum of equal speed is given by the equation

in which B is the collective weight of the centrifugal masses, and

D the load, suspended directly at D, to which the actual load is statically equivalent. The load D, and consequently the altitude

and the speed, can be varied at will, by shifting the weight L; which can be done either by hand or by the engine itself. The regulator may be acted on by the other end of the lever H K. The levers CD, CD should be horizontal when in their middle position; and then the ball-rods will slope at angles of 45°. Two

positions of the parts of the governor when the rods deviate from their middle position, are shown by dotted lines and accented letters. If convenient, the links E F, E F may be hung directly from the slides D, D.

The theory of this governor is illustrated by fig. 260. In any position of the parts, let A C be the axis of rotation; A B, a ball-rod carrying a ball at B; C, the point at which the lever C D = CA is jointed to the spindle; D, the central point of the slider at the end of that lever. About C draw the circle AD Q, cutting the axis of rotation in Q; join D Q; and draw DR and B P perpendicular to A Q.

Then when the position of the parts varies, and the speed is constant, the moment of the centrifugal force of the balls relatively to A varies proportionally to B P P A, and therefore proportionally to the area of the right-angled triangle A P B; and the moment relatively to A of the load which acts on the point D varies proportionally to D R, and therefore to the area of the right-angled triangle A D Q; but the areas of the triangles A B P and AD Q bear a constant ratio to each other-viz., that of A B2 to A Q2; therefore the moment of the centrifugal force at a constant speed, and the moment of load, bear a constant ratio to each other in all positions of the parts of the governor; and if they are equal in one position, they are equal in every position; and if unequal in one position, they are unequal in every position. Therefore the governor is truly isochronous.

To express algebraically the relations between the dimensions, the revolving mass, the load, and the speed; let B be the collective weight of the four balls; D, the total load which is actually or virtually applied at the points D, D; let the length of each ball-rod A B b; and let the length of each of the levers CD = C. In any position of the governor, let the angle QA B. Then, because A CD is an isosceles triangle, we have the angle QC D = 20. It is also evident that B P b sin ; A P = b cos ;

Ꭰ Ꭱ = c sin 2 = 2 c⋅ cos é sin .

=

Let n, as before, be the number of revolutions per second. Then the centrifugal moment of the balls relatively to A is

and the statical moment of the load relatively to A is

which two moments, being equated to each other, and common factors struck out, give the following equation:

364. Fluctuations of Isochronous Governors.—When a truly isochronous governor is rapid in its action on the regulator, and meets with little resistance from friction, it may sometimes happen that the momentum of the moving parts carries them beyond the position suited for producing the proper speed; so that a deviation from the proper speed takes place in the contrary direction to the previous deviation, followed by a change, in the contrary direction, in the position of the governor, which again is carried too far by momentum; and so on; the result being a series of periodical fluctuations in the speed of the engine. When this is found to occur, it may be prevented by the use of a piston working in an oil-cylinder or dash-pot; which will take away the momentum of the moving parts, and cause the regulating action of the governor to take place more slowly, without impairing its accuracy.

365. Balanced, or Spring Governors, (Silver's, Weir's, Hunt's, Sir W. Thomson's, &c.)—In this class of governors, often called Marine Governors, as being specially suited for use on board ship, the action of gravity on the balls is either self-balanced, or made, by rapid rotation, so small compared with the centrifugal force as to be unimportant. The centrifugal force is opposed by springs. To make such a governor isochronous, the springs ought to be so arranged as to make the elastic force exerted by them vary in the simple ratio of the distance from the centres of the balls to the axis.

In order that the action of gravity on the balls may be selfbalanced, if there are two balls only, they must move in opposite directions, in a plane perpendicular to the axis of rotation: which axis may have any position, but is usually horizontal. They might be guided by sliding on rods perpendicular to the spindle; but they are more frequently guided by combinations of linkwork, different forms of which are exemplified in Weir's governor and in Hunt's governor. If there are four balls, they are carried by a pair of arms like the letter X, as in fig. 259 (but with the spindle usually horizontal instead of vertical), and such is the arrangement in Silver's Marine Governor. The springs in balanced governors are seldom fitted up with a view to perfect isochronism; but for marine engines this is unimportant, as the principal object of applying governors to them is to prevent changes of speed so great

*It has been pointed out by Mr. Edmund Hunt that this form of governor is virtually a parabolic governor; for the common centre of gravity of the balls and of the load moves in a parabola, of a focal distance equal to half the altitude given by the formula.

and sudden as to be dangerous; such as those which tend to occur when the screw-propeller of a vessel pitching in a heavy sea is alternately lifted out of and plunged into the water.

Rules showing the relation between the deflection of a straight spring, or the extension of a spiral spring, and the elastic force exerted by the spring, have already been given in Article 342, page 386, and Article 345, page 389.

366. Disengagement-Governors.-The most complete example of a disengagement-governor is that commonly used for water-wheels, and sometimes also for

pendulums rotate. To make the slider rotate truly with the spindle, the part of the spindle on which it slides may either be made square, or may have a projecting longitudinal feather fitting easily a groove in the inside of the slider.

E is one end of a lever capable of turning about a vertical axis (not shown), and provided with a fork of four prongs, F, F, G, H. The prongs F, F are just far enough apart to clear the tooth D, as it sweeps round, when the spindle is turning at its proper speed, and the ball-rods and slider in their middle position; and the lever E is then in its middle position also. The prong G is below, and the prong H above, the level of the prongs F, F; and when the lever is in its middle position, the clear distance of G and H from the cylindrical surface of the slider B is one-half of the distance of F, F from that surface. When the spindle begins to fall below its proper speed, the slider moves downwards until the tooth D strikes the prong G, and drives the lever E to one side. Should the spindle begin to turn faster than the proper speed, the slider rises until the tooth D strikes the prong H, and drives the lever E to the contrary side. The lever E acts through any convenient train of mechanism upon the clutch of a set of reversing-gear, like the