1(XJ GOVERNORS. 



resistance M long ss the velocity is above its normal value, and reverses its 

 action when the velocity is below that value, the governor will bring the 

 velocity to the same normal value whatever 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 mechanism, 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. 



8. 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 stability 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 

 i' 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; and this is generally 

 expressed by the real root of the equation. If, by altering the adjustments of 

 the machine, its governing power is continually increased, there is generally ;i 

 limit at which the disturbance, instead of subsiding more rapidly, becomes an 

 oscillating and jerking motion, increasing in violence till it reaches the limit of 

 action of the governor. This takes place when the possible part of one of tin- 

 impossible roots becomes positive. The mathematical investigation of the motion 

 may be rendered practically useful by pointing out the remedy for these distur- 

 Itances. 



Tli is has been actually done in the case of a governor constructed by Mr 

 Fleeming Jenkin, with adjustments, by which the regulating power of the 



