maintained by Forces of Double Frequency. 507 



than the coefficient of the square of the velocity. When 

 the velocity is small the first term will be the dominating 

 term, and as the velocity increases, the second term will 

 increase more rapidly than the first and will become ulti- 

 mately the dominating quantity. Froude * has shown that 

 the forced oscillations of a ship rolling among waves are due 

 to a periodic forcing cause damped by resistance varying 

 with both the first and the second powers of the angular 

 velocity. Routh f has shown analytically that the period 

 of a pendulum in a very rare medium resisting partly as the 

 velocity and partly as the square of the velocity, is constant 

 throughout the motion and independent of the arc. Parker 

 Van Zandt J has shown experimentally that the free vibra- 

 tions of a system with a resisting torque due to both first, 

 and second power damping are isochronous. This law of 

 frictional force when introduced into the differential equa- 

 tion brings out all the above facts of experiment in double 

 frequency maintenance. 



The differential equation may be written as 



y + k'y ± hif r {n 2 — 2a sin 2pt + /3y 2 )y = 0. 



The sign + is introduced in the third term to indicate that 

 that part of the frictional force changes sign as velocity 

 changes sign. 



Assume as before, 



y = A 1 smpt-j-B ]L cosj)t + A 3 sin 3p£ + B 3 cos 3pt 

 + A 5 sin 5pt+B 5 cos 5pt + 



and consider in the limit all coefficients other than A x and 

 B 1 to be negligible. Collect the coefficients of sin pt 

 and cos pt with the aid of the following relations : 



y = A 1 sin pt + B 1 cos pt + .... 



y =2^(A 1 cosp^ — B x sin pi) + .... 



y sin 2pt = ^(A 1 cos pt + B 1 sin pt) -f- . . . . 



/=}A 1 (A 1 2 + B 1 2 )sin^+}B 1 (A 1 2 + B 1 2 )cos^+ .... 



The above are easily obtained by differentiation and trigono- 

 metric transformation. 



^ 2 =j D 2 (A 1 2 +B 1 2 ) cos* ( ,^ + tan" 1 B x /-A0". 



* Sir Phillip Watt's article on Shipbuilding- in the Encyclopedia 

 Britannica. 



t Routh's ' Advanced Rigid Dynamics/ Art. 364. 



X J. Parker Van Zandt, Physical Review, November 1917. 



