1893.] Phenomena somewhat analogous to Newton s Rings. 87 



Divisions of Theory. 



5. Imagine an electrical conductor ABCD (see fig. 1) consisting of 

 three parts, AB, BC, and CD, in each of which parts the electrostatic 

 capacity and other properties of the conductor remain constant, 

 those of the third part CD being precisely like those of the first part 

 AB, but the second part BC differing from the other two parts 

 either in its own dimensions or in the nature of the dielectric* by 

 which it is surrounded, or in both these respects. 



And consider an electrical wave passing along this conductor from 

 A towards D. Suppose its amplitude in the part AB is a, and that 

 from the point B (immediately after the instant of incidence of the 

 wave there and before any disturbance has reached C), a wave of 

 amplitude ab is reflected towards A, and a wave of amplitude ac 

 transmitted within the part BC towards C. The constants b and c 

 may be referred to as the coefficients of reflection and transmission re- 

 spectively. 



6. Then the elementary mathematical investigation of what may 

 be expected to occur falls naturally into three parts, namely : 



(1) The derivation of the coefficients of reflection and transmission 

 at the point B from the changes in the properties of the conductor 

 which occur there. 



(2) The relation between these coefficients and the similar ones 

 involved when the wave reaches C and thus encounters the reverse 

 change in the properties of the conductor. 



(3) The determination of the intensities of the total disturbances 

 reflected at or passing through B in the direction A and of those 

 transmitted through C in the direction D respectively, each being the 

 result of an infinite series of interfering waves produced by multiple 

 reflections within the part BC, these again being produced by the 

 original wave passing along AB. 



These three branches of the theory will now be taken in the above 

 order. 



7. I. Theory of single reflection and transmission of an electrical 

 wave along a conductor at a point where either its electrostatic capacity, 

 or its coefficient of self-induction, or both, change abruptly. Let the 

 following symbols be used, the same system, of units being under- 

 stood throughout : 



<p. Electrostatic potential. 



C. Electrostatic capacity per unit length of conductor. 



* It is to be distinctly understood throughout that the medium surrounding all 

 parts of the conductor is supposed to be a dielectric. All idea of its possessing 

 appreciable conductivity, and consequently absorbing a sensible portion of the 

 energy of the wave, is excluded from this theory. Also for all parts of the con- 

 ductor itself let 10 (the magnetic permeability) = unity. 



