Self-induction of Wires. 75 



upon the terminal apparatus, and F in the denominator is 

 defined by 



Umml=%- .- , 37^ y 7 =F(mZ), . . (6«) 



which is the determinantal equation arising out of the ter- 

 minal conditions 



V=Z Oat^=0, and V = Z 1 Catz = Z. . . (7e) 



[See equations (177) to (180), Part IV.] We have now to 

 add on to the numerator of A the terms corresponding to the 

 initial state of the terminal apparatus, when it is not then 

 neutral. As the process is the same at both ends of the line, 

 we may confine ourselves to the 2 = apparatus, according to 

 the figure. First we require the form of Z , the negative of 

 the generalized resistance of the terminal apparatus. It con- 

 sists of three parts, one due to the condenser, a second to the 

 primary coil, and a third to the presence of the secondary ; 

 thus, 



-Z =(K + S p)- 1 +(E 1 + L lP )-M^/(E 2 + L 2j p), . (8.) 



showing the three parts in the order stated. Now, as shown 

 in Part III., dZ /dp expresses twice the excess of the electric 

 over the magnetic energy in a normal system (when p be- 

 comes a constant), per unit square of current. Performing 

 the differentiation, we have 



tfZ _ S 2Wp L,My . 



dp - (K + IV) 2 Ul + B a + L 2P ( B, + L,pf ■ W) 



Here we may at once recognize that the first term represents 

 twice the electric energy of the condenser per unit square of 

 current, that the second term is the negative of twice the mag- 

 netic energy of the unit primary current, and that the fourth 

 is, similarly, the negative of twice the magnetic energy of the 

 secondary current per unit primary current ; whilst the third, 

 which at first sight appears anomalous, is the negative of twice 

 the mutual magnetic energy of the unit primary and corre- 

 sponding secondary current. Thus, if w be the normal cur- 

 rent-function, that is, by (4e), z%= — {m/R) cos 6, we have 



jr , s, — , w , and — ^ f , , . . (10e) 



as the expressions for the normal potential-difference of the 

 condenser, for the primary current, and for the secondary cur- 

 rent. If then V , C 1? and C 2 are the initial quite arbitrary 

 values of the difference of potential of the condenser, of the 



