PEIRCE. — OSCILLATIONS OF SWINGING BODIES. 67 



seems to be much more nearly proportional to the square of the angular 

 velocity. It will be convenient, therefore, to consider first the manner 

 in which the amplitude of an oscillat ng body would decrease if the 

 motion were resisted by a couple of moment proportional to the square 

 of the angular velocity. A roughly approximate solution of this prob- 

 lem was printed by Poisson in 1811, but is not accurate enough for 

 practical purposes. We shall do well to attack it in another way. 



If is the angular deviation in radians of the moving body from 

 the position of equilibrium, and b 2 the restoring moment, the mo- 

 ment of the couple due to the resistance of the air is of the form 

 2a(d0/dt) 2 ; and if K represents the moment of inertia of the swing- 

 ing system, the equation of motion is 



when the body is swinging in the positive direction. 



If for dd/dt we write &>, d 2 0/dt 2 is equal to w ■ dco/dd, and the equa- 

 tion becomes 



w • du> + (2 au> 2 + p20)d8 = 0, (7) 



which will become exact if we multiply through by e 4ad , so that, 



„ 2 = 2c .*-« + ^_^, ») 



or o) 2 = 2c e~ ke + m — mkO, (9) 



where c is a constant of integration. 



If — 0o is the value of the angular deviation at any elongation on the 

 negative side, and if 6\ is the next elongation on the positive side, then, 

 for the same value of c, 



2 c e k0 ° + m + mk$o = 0, (10) 



2 c e~ k0 i + m — mk$i = 0, (1 1) 



or (1 + k0 o ) e- ke ° =(1- k0 1 )e+ k \ (12) 



where k =4 a. This equation does not involve /3. 



For swings of large amplitude, it is easy to find Q\ graphically, 

 when k and O are given, by aid of this last equation. When O is small, 

 however, we may, in any practical case, develop each number of (12) 

 in a very convergent power series of which we need keep only terms 

 of order lower than the fourth. 



