592 Roj/al Astronomical Society, 



both sides of that position, and (so far as the action of those forces 

 affect it) will have no tendency to settle itself in the position of 

 equilibrium." This tlieorem supposes that some cause of disturbance 

 has once put the body into a state of oscillation ; and renders it 

 necessary to take account of such oscillation in j)lanning anjf me- 

 chanism which depends upon assuming the position of equilibrium 

 to be nearly preserved. 



If we examine the theory of the regulator, we shall see that the 

 friction which checks the motion takes place when the balls are 

 most distant from the axis, and (as the equable description of areas 

 is nearly observed) this occurs when the angular motion is least. 

 The whole maintaining force acts without check when the balls are 

 nearest to the axis, that is, when the angular motion is greatest. 

 Therefore, when the angular motion is least, the acting forces tend 

 still to diminish it ; when greatest, they tend to increase it. Hence 

 the inequalities of angular motion will increase till some new forces 

 come into play, which act in some different manner : and thus is 

 explained the obstinate adherence of the governor balls in some 

 cases to their elliptic motion. 



The author next proceeds to consider the ways in which an at- 

 tempt may be made to counteract the injurious effects of such oscil- 

 lations. These appear to be only two ; one, to make the oscilla- 

 tions of velocity much slower (or to make their periodic time 

 longer) ; the other, to make the oscillations quicker (or to make 

 their periodic time shorter). The first of these methods has the 

 effect of giving greater smoothness to the motion (an object of 

 great importance) ; and it is the principle which was adopted with 

 success in the clock-work of the Cambridge equatoreal. The se- 

 cond method endangers the smoothness of the motion ; but, as the 

 error has but a short time for accumulation, it ensures that the ob- 

 ject shall remain steady under the wire of the telescope far more 

 completely than the first. The construction attached to the clock- 

 work of the south equatoreal of the Royal Observatory is on this 

 principle; and it appears to answer extremely well. 



The mathematical problem proposed by the author in the present 

 communication is an investigation into the motion of governor balls, 

 for the purpose of deducing the time of rotation corresponding to a 

 p-iven expansion of the balls, and the periodic time of their oscilla- 

 tions, and the consequent oscillations in the angular speed of the 

 spindle ; and the subject is discussed on four different suppositions, 

 which, with their several principal results, are as follows : 1 . When 

 the balls are supposed to be acted upon by no force. The result 

 is, that the periodic time of oscillation is somewhat greater than 

 half the time of rotation. 2. When the axis which carries the balls 

 has a fly-wheel attached to it. In this case the periodic time of the 

 oscillations cannot be less than half the time of rotation, and may 

 be in any proportion greater. 3. When the balls are suspended by 

 rods from a horizontal arm carried by the regulator-spindle. The 

 result is, that the periodic time of the oscillations may be made 

 small in any proportion to the time of rotation. 4. On an assumed 



