380 On Forced Harmonic Oscillations of various Periods, 



It may be worth while to remark that in some cases, where we 

 cannot deal with phases, we are concerned principally with the 

 value of \/ (B/ 2 +^> 2 L/ 2 ), a quantity which practical electricians 

 are then tempted to call the resistance of the system. This 

 temptation should he overcome, and the name reserved for 

 B/, on which depends the amount of energy dissipated. It 

 must be admitted, however, that a name for V(E! 2 +p 2 U 2 ) 

 is badly required. Perhaps it might be called the "throttling/'' 



The corresponding theorem in cases when T vanishes is 

 deduced in a similar manner with use of the potential energy, 



y=i Cll 'f 1 2 + ic 22 '^ 2 2 + ic 3 3^3 2 +. . . . 



Thus, if we write -, , 



V^p'+L + Wm (22) 



we find 



c 2 2 r c 22 {c 22 2 +p% 2 2 )' v ' 



n f =b n ~x h ^\t { h f 2 ^X 12 l ' • • • ( 2 4) 



b 22 b 22 (c 22 2 +2> 2 b 22 2 ) 



As p 2 increases, the " stiffness " (represented by /j>') increases, 

 and the " resistance " diminishes. 



After what has been said it will not be necessary to occupy 

 space with illustrations of the present theorem. Indeed its 

 applications seem to afford less interest. It is curious that 

 here, again, the easiest examples would be taken from elec- 

 tricity, although the principle itself is one of general mechanics. 

 These (relating to the periodic charge and discharge of con- 

 densers through high resistances) may be left to the reader 

 who wishes to pursue the subject further. The application to 

 the theory of the conduction of heat may also be noticed. 



When the three functions T, F, and V are all sensible, it 

 is not generally possible to make the transformation to sums 

 of squares upon which our process was founded. There are, 

 however, special cases in which the same transformation 

 which is required to simplify T and V is successful also as 

 regards F. Among these are of course to be reckoned cases 

 in which F does not appear, and those where there is but 

 one other coordinate besides t^. Assuming that b 2S) b u , . . . 

 vanish, we have 



"*i 2 , • z ( c i2— P 2 <xu + ipb 12 ) 2 -. 



