./ c.isr.KM.i/.ATios or Tin: ri-.cii'koc.m. tiieorhm yi7 



III tin- appliiMtion of the precinling filiations In our prolilcm, it 

 will Ik- assuiiH'il that M is i-Ni-rywhero zero, so that 



- , 1 3P 



c dt 



k— 1 

 li will ill- .issimu-(l lu-llur that Ti = ctE and, siiux' P = £, 



4,r 



-i'-'-^H)' 



aiul is t iK-rt-forr a linc.ir fuiu-tion of E. a anfl k are in j^LMUTal poiiil 

 functions of tlie tncdiuni. The reason for setting M = (), is that it 

 apjx-ars essential to the following proof that U shall he linear in E; 

 that is, that the current density at an\- point he proportional to the 

 electric intensity.' 



With the foregoing ver>- brief review of the fundamental equations, 

 wi- are now prepared to pro\e the generalized rcciiirocal therorem. 

 Assuming a periodic stead>' state, so that 3 '9/ = ;w, we start with the 

 vector equation 



E = F--'-/l-v<i> (7) 



where 



* = /;exp(-'^r)pdv. 



Here F is the impressed intensity: that is, the electric intensity which 

 is not due to the currents and charges of the system itself. Also by 

 virtue of the assumption Af =0, 



whence (7) can be written as 



-U + -J ^.^-^r)u.W=rp, (8) 



where G = F — '^•i>. 



' The c|uestion as to whether the generalized theorem itself, and not merely the 

 foregoing proof, is rcstrittefl in general to the case where Af is everywhere zero has 

 not as yet received a conclusive answer. There are reasons, however, which cannot 

 be fully entered into here, which make it appear probable that the theorem itself 

 is in general restricted to the case where the current density contributing to the 

 retarded vector potential is linear in the electric intensity and the two vectors are 

 parallel. Subject to the hy[x)thcsis and assumptions of quasi-stationary systems, 

 however, the restriction Af =0 is not necessary-. The writer hopes to deal with these 

 questions in a future paper. 



