386 Royal Society : — 



In one class of regulators of machinery, which we may call modera- 

 tors # , the resistance is increased by a quantity depending on the ve- 

 locity. . Thus in some pieces of clockwork the moderator consists of 

 a conical pendulum revolving within a circular case. When the ve- 

 locity increases, the ball of the pendulum presses against the inside 

 of the case, and the friction checks the increase of velocity. 



In Watt's governor for steam-engines the arms open outwards, 

 and so contract the aperture of the steam-valve. 



In a water-break invented by Professor J. Thomson, when the 

 velocity is increased, water is centrifugally pumped up, and overflows 

 with a great velocity, and the work is spent in lifting and communi- 

 cating this velocity to the water. 



In all these contrivances an increase of driving-power produces an 

 increase of velocity, though a much smaller increase than would be 

 produced without the moderator. 



But if the part acted on by centrifugal force, instead of acting 

 directly on the machine, sets in motion a contrivance which conti- 

 nually increases the resistance as long as the velocity is above its nor- 

 mal value, and reverses its action when the velocity is below that value, 

 the governor will bring the velocity to the same normal value what- 

 ever variation (within the working limits of the machine) be made 

 in the driving-power or the resistance. 



I propose at present, without entering into any details of mecha- 

 nism, to direct the attention of engineers and mathematicians to 

 the dynamical theory of such governors. 



It will be seen that the motion of a machine with its governor 

 consists in general of a uniform motion, combined with a disturbance 

 which may be expressed as the sum of several component motions. 

 These components may be of four different kinds :— 



1 . The disturbance may continually increase. 



2. It may continually diminish. 



3. It may be an oscillation of continually increasing amplitude. 



4. It may be an oscillation of continually decreasing amplitude. 



The first and third cases are evidently inconsistent with the sta- 

 bility of the motion ; and the second and fourth alone are admissible 

 in a good governor. This condition is mathematically equivalent to 

 the condition that all the possible roots, and all the possible parts of 

 the impossible roots, of a certain equation shall be negative. 



I have not been able completely to determine these conditions for 

 equations of a higher degree than the third ; but I hope that the 

 subject will obtain the attention of mathematicians. 



The actual motions corresponding to these impossible roots are 

 not generally taken notice of by the inventors of such machines, who 

 naturally confine their attention to the way in which it is designed 

 to act j and this is generally expressed by the real root of the equa- 

 tion. If, by altering the adjustments of the machine, its governing 

 power is continually increased, there is generally a limit at which 

 the disturbance, instead of subsiding more rapidly becomes an os- 

 cillating and jerking motion, increasing in violence till it reaches the 

 * See Mr. C. W. Siemens " On Uniform Rotation," Phil. Trans. 1866, p. 657. 



