Vibrations elucidated by Simple Experiments. 249 



In the same notation, (11) may be written 



ffV=(l-7W (16) 



Eliminating a between (15) and (16), we obtain the 

 biquadratic in t, 



T*-{fJL 2 + V 2 )T 2 + (l-V 2 )lM 2 V 2 = 0. . . . (17) 



Thus solving for t 2 , we have 



2t 2 = fi 2 +v 2 ±{(fu 2 -v 2 y + ±rffj?v 2 }\ . . (18) 



The two values of r 2 here shown may be called t 2 and a 2 . 

 Accordingly we see that the periods, a and t, of the coupled 

 vibrations differ from each other and from those, /ul and v, of 

 the vibrations of the separate systems, and also that the 

 difference between <r and t exceeds that between /jl and v. 

 Further, even when jjl and v are alike, a and t still differ 

 from each other and from jjl (for all finite values of the 

 coupling 7). 



As to special cases, we may note the following : — 



For v=fi, we find 



T = fju\/(l + r / ) and o- = yuV(l — 7). . . (19) 

 For v = fi and 7 very small, this gives 



T = 0? = fl (20) 



For 7=1 but fi unlike v, we find 



t 2 = /x 2 + ^ 2 and <7 2 = (21) 



Initial Charge. — If one capacity is initially charged, the 

 other uncharged, and both currents zero, which is the usual 

 case, we may write 



y=0, z=b, J=0, *Uo, for t = 0. . (21a) 



Inserting these conditions in (13) and (14) and in the 

 differentiations of these with respect to the time, we find the 

 following special solutions : — 



^ = MR -^-51 6 cos pt-M.Il -£^-\b cos qt, . (21b) 



