M. G. Kirchhoff on the Motion of Electricity in Wires. 401 



Through considerations of the same kind it will be seen that the 

 equation (3) receives a similar form ; thus we have 



w=:2i\os—. 

 u 



These values of V and w are to be substituted in equation (2) ; 

 when this is done, setting, for the sake of brevity, 



and denoting the resistance of the entire wire, that is, the quantity 



/ 



by r, we obtain 



, I (^e 4 ^i\ 



From this equation, in connexion with equation (4), viz. 

 ^■__^ 



we have to determine i and e as functions of s and t. 



A partial solution of the differential equation is found by 

 setting 



e = X sin ?w 



i=Ycosrw, 



where n denotes an arbitrary constant, and X and Y are unknown 

 functions of t. By this the equations become 



Y= 



. / / „ 4 ^\ 



From this we obtain, by eliminating Y, 



rf^X cV «?X ^^^^ X — (\ 



IF^To^ilt'^T 



The general integral of this differential equation is 



X = C,e-^i' + C2e->^2' 



where C, and C^ are two arbitrary constants, e the basis of the 

 hyperbolic logarithms, and A-, and Xj the roots of the quadratic 

 equation 



