502 On Resistance and Conductance Operators. 



any other part of the system than where the impressed force 

 is, it is necessary to know the connexions, so that we may 

 know the ratio of the current in a normal system at the new 

 place to that at the old ; inserting this ratio in the summa- 

 tion, and modifying the external Z to suit the new place, 

 furnishes the complete solution there. Or, use the more 

 general resistance-operator Z xy , such that e x — Z xy O y , connect- 

 ing the impressed force at any place x with the current at 

 another place y. 



18. When the initial current is zero, as happens when there 

 is self-induction without permittance at the place of e, and in 

 other cases, (54) gives 



1 1 

 Zo = ^^W'> (55 ) 



showing that the normal systems may be imagined to be 

 arranged in parallel, the resistance of any one being (—pZ 1 ). 

 To express the impulsive inductance Z ' in terms of the 

 normal Z's, multiply (54) by e and take the complete time- 

 integral. We obtain 



J,(C-|-)A=2(U-T)=-.2^, . . (56) 

 remembering (29). Or, using (26), 



^'=2^2, (57) 



In electrostatic problems the roots of Z = are real and 

 negative, as is also the case in electromagnetic problems. 

 There are never any oscillatory results in either case, and the 

 vanishing of 71 is then accompanied by vanishing of the cor- 

 responding normal functions, to prevent the oscillations which 

 seem on the verge of occurring by the repetition of a root 

 which Z' = implies. When both energies are present, the 

 real parts of the imaginary roots are always compelled to be 

 negative by the positivity of U, T, and of Q the dissipativity. 



When Z is irrational, it is probable that the complete solu- 

 tion corresponding to (54) might be immediately derived 

 from Z. In the case of (41), however, the application is not 

 obvious, although there is no difficulty in passing from the 

 (54) solution to the corresponding definite integrals which 

 arise when the length of the circuit is infinitely increased. 



October 15, 1887. 



* In Part III. of "On the Self-induction of Wires" I employed the 

 Condenser Method, with application to a special hind of combination ; 

 but, as we have seen from the above proof, (54) is true for any electro- 

 static and electromagnetic combination provided it be finite. 



