244 A IT. P>. Hopkiuson. [June Hi. 



resisted by a force proportional to the excess of the actual velocity 

 over that of the steady motion, or by a true viscous force, and the 

 magnitude of the force is, in the simple case when p = 0, 



Taking this force into consideration, as well as the similar force 

 of opposite sign whose existence was proved in the first paragraph, 

 we find finally that the free oscillations of the system are given by the 

 equation : 



where y and y' have the following values nearly, if p is small, 



y , 



These results have been obtained for a single-phase motor, with 

 a constant self-induction and a sine-wave E.M.F. A generator running 

 in parallel with a number of others is, of course, covered by the 

 same equations. The results are applicable, with slight modification, 

 to a two-phase or three-phase machine, and it would be possible, if 

 worth while, to find the alterations introduced by the varying self- 

 induction and distorted wave-forms which exist more or less in all 

 actual dynamos. In no case, however, could accurate quantitative 

 results be arrived at without great labour, for the forces here 

 investigated are small, and in actual work a great many small dis- 

 turbing causes would have to be taken into account (such, for example, 

 as small changes in the resistance overcome by the motor)* before an 

 accurate quantitative criterion of stability could be arrived at. A 

 good deal of valuable information can, however, be got out of the 

 equations as to the general effect upon stability of running of varying 

 the constants of the machine. 



(1) y' diminishes as 8 increases or the damping increases with the period 

 of the oscillation. Hence increased fly-wheel effect always results in 

 better damping owing to the increase in the period. Increasing the 

 self-induction has the same effect; arid it also works in favour of 

 stability by diminishing y. A mere alteration of M without altering 8 

 has no effect because it alters y and y' in the same ratio. 



(2) The damping is proportional to * /3 Q 2. Now the steady torque is 



* A case of great practical importance, in which the changes of load can be 

 calculated easily, is that of the rotary converter. 



