238 Mr. B. Hopkinson. [June 16, 



compared with 2ir/p the period of the alternating current. The 

 external E.M.F. remains the same E cospt + F sin pt the induction 

 A sin 6 is also undisturbed by the oscillation. 



Forming the equation for the E.M.F. at the terminals of the motor 

 in the usual way and equating it to the impressed E.M.F. we find : 



n^ = pu + Lu+-=- 



= P {(^o + a) sin pt + (/3 Q + /?) cos pt} 

 + Lp {(a + a) cos pt - (/3 + P) sin pt)} 

 + La sin.pl + L/5 cospt 



This equation is rigorously accurate under the assumptions proposed. 

 Now equate the coefficients of cos pt and sin pt on the two sides, neglect 

 products of the small quantities a, /3, , a, etc., and separate the large terms 

 corresponding to steady motion and the small terms corresponding to 

 disturbed motion in the usual way. 



Thus for steady motion 



and for the disturbed motion 



= ..................... (1), 



= ..................... (2). 



The torque developed by the motor is 



u j- = A cos B {(a + a) sinpt + (/3 + ft) cospt} 



= A cos (pt + ) {(a + a) sin pt + (/? -f /?) cos pt} 



= (A/? + A/3- Aa ) + terms of period ir/p and quicker 

 period, 



again neglecting products of the small quantities. The large part of 

 this, JA/2 , is equal to the constant resistance, the small balance is 

 available for accelerating the motor. Thus the third equation is 

 obtained : 



..................... (3). 



To solve equations (1), (2), and (3), we write as usual a = I*e xt , 

 fi = Qe xt , and = TLe xt , and so get, after substitution and elimination 

 of P, Q, and K, 



