GOVERNORS 331 
neasured from the centre of the ball A to the axis of the joint at B. 
e total weight of the arm is aw= W,, and the centre of gravity of the 
m will be taken at the middle of the length a. O is the point where 
- the axis of the arm AB intersects the vertical axis of revolution OY. 
In Fig. 515, B and O coincide. In Figs. 516 and 517, the joint B is 
on an arm fixed to the vertical spindle, the axis of the joint B being at 
horizontal distance c from OY. In Fig. 515, ¢ is therefore =0. 
For simplicity, in what follows attention will be directed in the first 
“instance to Fig. 516. 
Consider an indefinitely small length dz of the arm at a distance 
ra from B. The centrifugal force df of this small length of arm is 
; Beets +c) , where @ is the inclination of AB to OY. The moment 
of this aaeitieal force about B is 
. woos Af = neer(e sin 8+ cle 00s Owe ooo (sin i+ In), 
and the resultant moment about B of the centrifugal force of the whole 
is 
ww? cos 6 
ww? COs (F : a 
(sin 0 22de-+c{ ztn) =" 3 sin +> 
a i cos 4¢ ies 4: :) 
. % g 3 2/ 
_ Considering now all the forces acting on A and AB, and taking moments 
— about B, 
wet (h— cot 0) + 0) + wietacos(§ sin 0 + s)= W(r- e)+W, >. 
ti aati g cot 0=", sind ="—*, and 00s 6 = Mr= 0), the equation of 
eae eetim reduces to (W147) }= =W+, To make this 
= apply to Fig. 517, it is only necessary to change ¢ to —c. Hence the 
) - general equation is om {W+ "(1 +2)\ = we, 
we 
If c=0 (Fig. 515), then *(w 4.%3)- w+ 44, andh— 2 2 
. ’ g 3 2 wet w* 
3 
Since . will generally be comparatively small, the equations for Fig. 
«BIS may - taken as applying to Figs. 516 and 517 also. 
J Tt will be seen that the effect of the mass of the arm AB is equivalent 
_ to increasing the centrifugal force of A by an amount due to an increase 
in its weight of M4 , and increasing the downward pull at A by an amount 
equal to 1, 
