PEIRCE. — OSCILLATIONS OF SWINGING BODIES. 69 



If we differentiate (6) with respect to t, and represent dw/dt by r, 

 we shall get the equation 



rdr 

 4 ar + fi 2 

 or 



+ uxlu = 0, (19) 



r _|l % ^ + |!) = C-2^ ) (20 ) 



and C, the constant of integration, may be determined from a considera- 

 tion of the fact that when w — 0, 6 is — 6q. 



If a swinging system oscillates about a position of equilibrium under 

 the action of a righting moment proportional to the deviation and a 

 resisting couple proportional during the whole motion to the first power 

 of the instantaneous angular velocity, the equation of motion has the 

 familiar form 



If p2 = @2 _ a 2 f an d if m anc J n are the roots of the equation 



x 2 + 2 ax + £2 = 0, (22) 



m = — a + pi, n — — a — pi, 



and we have 6 = e~ at (L cos pt + M sin pt), (23) 



or = Ae~ at sin (pt — e), (24) 



where A and e are constants of integration. If, using t and 6 as co- 

 ordinates, we plot (24), it is clear that the curve 6 = Ae~ at touches 

 the curve 6 —Ae~ al sin (pt — e) when pt — e = (2 k + %)tt, so that if 

 the time be counted from the date of one of these points of tangency, the 

 corresponding solution of (21) may be written in the form 



6 = Bc~ at cos pt. (25) 



The complete period of the oscillation (T) is 2ir/p. The ratio of the 

 amplitudes at two consecutive elongations is e a7T p and the logarithmic 

 decrement is airjp. The ratio of the amplitudes at two consecutive 

 elongations on the same side of the position of equilibrium is e 2a7T/p , 

 and we have 



a = 2X/T, (3 2 = 4 (tt2 + a2)/T 2 . (26) 



The maxima of the curve (24) occur at times defined by the equation 

 tan (pt — e) = p/a ; or sin (pt — e) = p/(3, 



