240 Mr. B. Hopkinson. [June 16, 



whence , M _ A 2 /o + (Aa - 2M8 2 ) 2L P 



My = 2 



/KX 



If, therefore, L^>/3, as is nearly always the case, 7 is positive, and 

 the motion is unstable, the oscillations if once started continually 

 increasing in amplitude according to the law ei l . 



In the above it has been assumed that the field is unaffected by the 

 oscillations, remaining the same in the disturbed as in the steady 

 motion. As a matter of fact this is not usually the case, the field is 

 disturbed by the oscillation owing to the varying armature reaction. 

 The general effect is easily expressed thus. The induction linked with 

 armature and field coils in the steady motion is supposed, as before, to 

 be A sin 0. In the disturbed motion the field is slightly altered in 

 .amount, and slightly distorted, in a periodic way, and becomes 

 I = (A + a) sin + b cos 0, where a and b are small quantities depen- 

 dent on a, /?, and . Terms involving sin 20 and cos 20 will also 

 appear to some extent, but may for practical purposes be neglected. 

 The current in the armature is 



u = (a + a) sin pt + (/? + P) cos pt 



= a sin 6 + j8 cos 6 



+ ( a + P<) sin + (p - ao ) cos 0, 



.and the general effect of the varying current on the field is that 

 a = K( a + /3o), b = X 08 -orf), 



where K and X are constant for any given state of steady motion. K 

 represents the change in total induction produced by varying the 

 " wattless " component of the current, and X the change in distortion 

 produced by varying the other component. Taking these expressions 

 for the induction, it is easy to find the time of oscillation and rate of 

 increase as in the simpler case investigated above. It will suffice here 

 to give the results. The period of the oscillation is 27T/S, where 



and the amplitude increases at the rate ey { , where 



^ A - Xa > t ( L 



- P* {(A -*)* + **#,'} ] ......... (6). 



