64 Messrs. Lodge and Howard on Electric Radiation. 

 Integrating with regard to a, this becomes 



Jo s/(b 2 + C*)-b 



where numerator and denominator are of the same form if the 

 limits for the numerator portion have Z subtracted from them 

 both. Performing the integration of the two parts separately, 

 and simplifying, we get 



It may be worth while to write down the form this assume 

 when c is moderately small compared with Z, viz. 



If we now put for c the geometric mean distance of the 

 points in a cross section of a rod of thickness d, we shall have 

 the mutual induction of the parts of all the filaments in that 

 rod upon each other, L e. the self-induction coefficient of the 

 rod. And unless the rod is very short and thick, it will be 

 permissible to neglect the c/l terms. 



Now the geometric mean distance of the points in a circular 

 section varies from -Jd, when they are concentrated into its 

 circumference, to ^e~*d, or *3894 d, when they are spread 

 uniformly all over it. The first case corresponds to our 

 rapidly periodic currents, and gives, as the self-induction of a 

 rod in which currents keep to the periphery, 



L = «(log£-l)s 



whereas if the currents penetrate all through its section, by 

 reason of being of slowly changing strength, 



L = 2Z| 



0«4-»). 



The difference is not marked : at least for the case supposed, 



of non-magnetic material. 



Hertz employed this last formula, quoting it apparently 



from Neumann ; but he says that in Maxwell's theory the j 



turns into £. We do not know how he makes this out, but 



suppose he is somehow right ; and it is this uncertainty which 



has caused us to refrain from going into minutise on the sub- 



41 

 ject, and to be satisfied with using merely log -y, instead of 



