Inductance of a Wire to an Oscillatory Discharge. 439 



which is positive for all values of k from to go , hence K 

 increases continuously with k. For & = 0, this becomes 



'dK\ 



. dk A =0 ~~ 



7T 



sin- 

 4 



~V2 



which assists in plotting K as a function of k. 

 Differentiating again, we obtain 



dk 2 



, (7 + 3*«)cos-g + 2*sin 5 } 



6 



(29) 



(30) 



Since this expression is positive for all values of k from 

 to oo , we see that K plotted as a function of k is a curve 

 which is always convex to the axis of k. Thus the nature of 

 R'yR' as a function of k is sufficiently determined. 



Pairs of corresponding values of K and k for a few typical 

 cases are shown in the accompanying Table, and part of the 

 curve coordinating them is given in fig. 1. It is not necessary 

 to plot much of the curve, as only a small part of it can 

 apply to any actual case. For, although k may have any 

 positive value up to go , the high values of k, as already men- 

 tioned, correspond to low values of p and so exclude them 

 from the application of the high-frequency formula. 



Table showing the values of K = R"/B/, the ratio of equiva- 

 lent resistances to waves with damping and without. 



Damping Factor, 

 h = cot 0. 



Subsidiary quantities involved. 



Ratio of 

 Resistances 

 K=^R fR . 



0/2. 



6 - 2 = l-fP. 







45° 



1 



1 



i- =0-0798 



4tt 



42° 44' 



1-006362 



T044 nearly 



J* =00955 



107T 



42° 16' 



1-00913 



1-054 „ 



i =01595 

 2ir 



40° 28' 



102614 



1097 „ 



— =0-319 



7T 



36° 9' 



11018 



1-228 „ 



1 



22° 30' 



2 



2-197 „ 



2 



13° 17' 



5 



4 602 „ 



3 



9° 13' 



10 



7-85 



2 H 2 



