124 PROCEEDINGS OF THE AMERICAN ACADEMY. 



where the integral is again taken over the roundabout path of Figure 

 2. We may write 



r _ 4 udu udu 



u 



u du 

 J ° U 2^ + 2^ + ^' 



where the first integral on the right is taken along the junction line, 

 and the second about u = i — |k. The former is approximately 



XI u (I + \ V 1 + 1) du = \ XI " + viJ +^ du - 



which is of order k 2 and vanishes to the present degree of approxima- 

 tion. The latter is 



-27H 



1 



u 



1 Kl 



U+ I — ~K — 



2 u 



- 2iri 



2 K 



U = I 



l 2i (1 + Ik) 



= — 7Tt — - 7TK. 



The value of X is therefore, 



A^ = -a e- hwK = - a (1 - |tt/c). 



The point d'arret therefore lies slightly nearer the origin than the point 

 of rest. 



To obtain the amount of energy lost by radiation during the swing 

 from the starting point of no velocity to the point d'arret we might 

 evaluate the integral of K(d' 2 x/d.T 2 ) 2 as we have evaluated the integrals 

 for the time and displacement. But it is simpler to figure the final 

 kinetic energy and potential energy and to subtract their sum from the 

 initial potential energy %a 2 . The final velocity is v — 4k.i', with 

 x = X = — a(l —^tk). The kinetic energy \v 2 is therefore negligible 

 to the order of our approximations. The potential energy is \x 2 = 

 \{a 2 — ttk) which is less than ^a 2 by the amount §7tk. Thus the 

 energy radiated during the swing is essentially the same as that 

 radiated during one swing on the assumption that the motion is 

 practically simple harmonic. 



