340 APPLIED MECHANICS 
Hence the area of the rectangle H,K,K,H, is equal to the area of the 
figure aF', F,A,. 
Now suppose that when the ball is at A, the speed suddenly increases 
‘to N,, and suppose that the ball is preveiited from rising by a force § 
acting vertically downwards at A,. The centrifugal force will now be 
aF,=cwNzr,. Make K,L,=§, den H,L,=w+S8, the total downward 
force at A,. For equilibrium it is obvious that aF, x OH, = H,L, x A,H,. 
Next suppose that the force § is diminished so as “to allow the ball to rise 
to A,, then remembering that the speed during this change is N,, the 
centrifugal force will be directly proportional to the radius, and will 
therefore be represented by the ordinates of the straight line F,F,, 
which when produced passes through H,. In order that there may be 
equilibrium in each position of the ball as it rises Fh=(w+s)r, where s 
is the vertical effort at the centre of the ball when its distance from the 
w+s 
h 
and w+s will be represented by the abscisse of the straight line L,K,, 
which when produced passes through O. The work done on the force s 
as the ball rises from A, to A, is therefore represented by the area of the 
triangle K,L,K,. In the same time the work done by the centrifugal 
force is represented by the area of the figure aF,F,A,, but the part of 
this, aF,\F,A,, represents the work done in raising w, therefore the 
external ‘work done is represented by the area F\F,F;. 
If straight lines OL,K, and H,F,\F, be drawn, it is easy to show 
that the external work which the governor is capable of doing as the ball 
descends from A, to A, is represented by either the area of the triangle 
K,L,K, or the area F,\F,F,. 
If H,H, is the maximum or total lift of the balls, then for each ball 
the maximum power of the governor is represented by the area of the 
triangle K,L,K, when the ball is ascending, 
and by the area of the triangle K,L,K, when 
the ball is descending. But if the ascent from 
H, to H, is made in three steps (Fig. 529) 
instead of one, the external work done for each 
ball will only be that represented by the sum of 41 Ky Hy 
the areas of the triangles shaded with vertical Fig. 599. 
lines, and if the descent from H, to H, is made 
in three steps instead of one, ‘the external work done for each ball 
will only be that represented by the sum of the areas of the triangles 
shaded with horizontal lines. 
The horizontal widths of the triangles K,L,K, and K,L,K, at any 
given level measures the vertical effort s for each ball of the governor at 
the centre of the ball at that level, the width of the triangle K,L,K, 
being the effort during ascent, and the width of the triangle K,L,K, 
being the effort during descent. The effort at the sleeve is got by multi- 
plying the effort at the balls by the ratio of the vertical motion of the 
balls to that of the sleeve. 
For the direct loaded governor (Fig. 521), the length H,K, (Fig. 528) 
W 
eee § 
, but z is constant, therefore US constant, 
AES F 
axis is 7, hence — = 
r 
is made equal to 
