

Criterion of Potential Energy. 701 



24. The collar K can partake of the axial motion of the 

 block C, as the latter slides lengthwise of the axis A A; K is 

 incapable of rotating with respect to the frame B, but is so 

 connected to G as not to interfere with C's rotation. The 

 parts in the second governor are indicated by accented letters, 

 and are in all respects equal and similar to those in the first 

 governor. The guides Q Q are integral with the frame B 

 on the one hand, and with the inner frame P on the other 

 hand, and upon these guides the blocks P P can slide without 

 rotation. The pieces P K P'K' are connected up as shown 

 by the four links MMM'M', which are equivalent to a 

 jointed parallelogram. For the present we leave out of 

 consideration the mitre-gears S S T T', to which reference 

 will be made later. 



25. It is evident that, in addition to any freedom of motion 

 which the frame B may have, and exclusive of the rotational 

 motions of the governors, the system made up of the two 

 governors and the parallelogram linkwork has one deo-ree of 

 freedom, corresponding to a motion of each of the blocks P P 

 (say) towards the centre of the framework, with simultaneous 

 motion of K K' away from the centre, and increase of the 

 angle 6 which the arm E H makes with the axis A A.'. This 

 angle 6 may conveniently be taken as the coordinate corre- 

 sponding to the freedom just defined. For the remaining 

 coordinates of the system we may take x, y z z, the Cartesian 

 coordinates of the centroid, and three angular coordinates 

 denning the orientation of the frame B, together with the 

 angles %, %' (measured from a standard configuration) through 

 which the respective governors E H G, E'H / G / have turned 

 with respect to the frame B. 



26. Let the system, with the exception of the rotatable 

 governors, have the moment of inertia I 2 about the axis A A' 

 and at any instant (t) let the frame B be turning about the 

 axis A A' with angular velocity g>\. Let I 2 be the moment of 

 inertia of the whole system including the governors about the 

 axis Q Q; and co 2 the angular velocity of the frame B about 

 the instantaneous axis QQ; I 3 , co 3 being the correspondino- 

 quantities referred to an axis through the centroid of the 

 system perpendicular to the plane of fig. 2. If M is the 

 mass of the entire system, and if the moment of inertia of 

 either governor about its mechanical axis, the kinetic energy 

 may be written 



T = I M(.? + if + ?) + % W -f i W + 1 I,*/ + 1 I Q 02 



+ ^t(% + a) 1 )- + AHr + W] ) 2 



27. Here I is a quantity of the same nature as a 



• (22) 



moment 



