1903.] The " Hunting " of Alternating- Current Machines. 



Lx + p ~Lp -Ap 



Lp Lz + p Aa? 



A 



= 0, 



which reduces to 



A 2 L(z 2 +^? 2 ) + A *px = ... (4) 



The condition of stability of motion is, as explained in works on 

 dynamics, that the real roots of this equation shall be negative, and 

 that the real parts of the complex roots shall be negative. The 

 criterion for this can be written down,* but in this case it is simpler to 

 proceed by approximation. It is known that # is a small quantity, the 

 period of the oscillation we are investigating being in any practical 

 case at least fifteen times that of the alternating current. Neglect 

 x/p and xp/Lp* therefore altogether in the first instance. Thus we 

 obtain 



(Aa + 2Mz 2 )(Ly + /o 2 ) + A 2 V = 0, 

 whence 



so that to this order of approximation the motor executes simple 

 harmonic oscillations of constant amplitude and period 2ir/8. This, 

 subject to an approximation which holds good in practical cases, is the 

 result obtained by Kapp. Now, suppose that x = y + i8, and sub- 

 stitute in the original equation (4). Neglect y 2 , 2 /p 2 , and y/p, but 

 keep x/p, xp/Lp 2 , etc. 

 The result is 



4Myt8 (L 2 / + /> 2 ) + (Aa - 2M8 2 ) 2LiSp + A 2 pi8 = 0, 



* The condition that the real roots and the real parts of the complex roots of 

 the biquadratic 



ax 4 + bx % + cy? + dx + e = 



shall all be negative is that the quantity 



and the coefficients a, b, c, d, e shall all have the same sign (see Kouth's * Eigid 

 Dynamics' (1892), vol. 2, p. 192). In the case of the biquadratic here treated, 

 this condition reduces to 



2A(2Lao + A)-?^(IV-p2) > 0, 



which is equivalent to the condition Lp < p found later on, if, as is the fact, the 

 first term can be neglected in comparison with the second. 



