Oscillations of various Periods. 377 



Thus, 



F =* R (RTs^ + ^) 2+ * s (ra^-^) 2 



and if L, M, N" be the induction-coefficients of the two con- 

 ductors, 



w . • 9 LS 2 + 2MSR + NR 2 



T=i ^ (E+Sp 



■ : (L-M)S + (M-N)R , i; „ T 9M , N , 



Accordingly , 



LS 2 + 2MSR + NR 2 _ (L-M)S + (M-N)R 



(R + S) 2 > ai2_ R+S 



a 22 = L — 2M + N; 



, &i2 = 0, & 22 = R + S; 



11 R-fS 

 and thus by (6), (10), 



_SR _ £ 2 _ {(L-M)S + (M-N)B}» ^ 



R+S^R+S (R + S^+^L-^M + N) 2 ' * l ; 



T ,_ LS 2 + 2MSR + NR 2 _ {(L-M)S + (M-N)BP 



(R+8) 2 (R+S) 2 (L~2M + N) 



{ ( L-M)S+(M-N)R}' ( 



(L-2M-fN){(R + S) 2 +p 2 (L-2M + N) 2 } ' k j * 



It should be remarked that (L — 2M + N) is necessarily 

 positive, representing twice the kinetic energy of the system 

 when the current in the first conductor is -f 1 and in the 

 second —1. 



Of the three terms in (14) the second and third cancel one 

 another when p vanishes, and when p is very great the third 

 term tends to disappear. The first and second terms together 

 may be put into the form 



LN— M 2 

 L-2M + N' ( 15 ) 



independent (as it should be) of the resistances. In this 

 (LN— M 2 ) is necessarily positive, bat may be relatively small 

 when the wires are wound together. The energy of the 

 system is then very small, when the currents are so rapid 

 that their distribution is determined by induction. 



