556-560] Oscillations in a Network 487 







dx 



current be measured towards the plate, so that the relations ^ = -^ , etc. 



will still hold. Let conductor 1 contain an electromotive force E l and be 

 of resistance R^. 



The quantities as lj # 2 > ... may be taken as Lagrangian coordinates, but 

 they are not, in general, independent coordinates. If any number of the 

 conductors, say 2, 3, ... s meet in a point, the condition for no accumulation 

 of electricity at the point is, by Kirchhoff's first law, 



from which we find that variations in # 2 , ac s , ... are connected by the 

 relations 



&C 2 &E 3 ... &C g =0. 



Let us suppose that there are m junctions. The corresponding con- 

 straints on the values of &EJ, &r a , ... can be expressed by m equations of 

 the form 



!&! -f Oa&tfa + + G&W&CM = j 



> .................. (518), 



M<fei + M# 2 + ... 4 &n&** = ) 



etc., in which each of the coefficients Oj, a 2 , ... a n , 61,- ... has for its value 

 either 0, +1 or 1. 



The kinetic energy T will be a quadratic function of d! lt 2 > etc., while 

 the potential energy W (arising from the charges, if any, on the condensers) 

 will be a quadratic function of oc l , a? 2 , .... The dynamical equations are now 

 n in number, these being of the form (cf. equations (509)), 



These equations, together with the m equations obtained by applying 

 Kirchhoff's first law to the different junctions, form a system of m + n equa- 

 tions, from which we can eliminate the m multipliers X, /, ..., and then 

 determine the n variables x lt # 2 , ... x n . 



560. As an example of the use of these equations, let us imagine that 

 a current / arrives at A and divides into two parts i l} i 2 , which flow along 



arms AGE, ADB and reunite at B. Neglecting induction between these 

 arms and the leads to A and B, we may suppose that the part of the kinetic 

 energy which involves i\ and i" 2 is 



