On the Passage of Electric Waves through Tubes. 125 



When p = 3 there are 32 such groups. They occur as 

 follows : — 



Degree of groups 4 6 8 12 24 



Number „ 1 3 3 10 15 



When p = l there are 22 such groups. They occur as 

 follows : — 



Degree of groups 8 14 28 56 



Number „ 1 1 7 13 



When p — 1 is divisible by 8 there are 27 such groups. 

 They occur as follows : — 



Degree of groups p 2p 4p Sp 



Number „ 12 9 15 



When p — 1 is divisible by 4 but not by 8 there are 23 

 such groups. They occur as follows — 



Degree of groups 2p 4/> 8p 



Number „ 1 8 14 



Whenp — 1 is not divisible by 4 andp does not have one 

 of the three values 2, 3, 7, there are 18 such groups. They 

 occur as follows : — 



Degree of groups 4p Sp 



Number , 6 12 



jj 



The three groups whose degrees are 4, 8, and p respectively 

 are primitive. All the others are nonprimitive. When 

 p = 2 there are five commutative groups, but when p > 2 there 

 are only three such groups. When p = 2 there are three 

 non-commutative groups that are not simply isomorphic to 

 any non-regular transitive group. When p — 1 is divisible 

 by 4 there are five such groups. When this condition is not 

 satisfied and p> 2 there are four such groups. 



Paris, December 1896. 



XVIII. On the Passage of Electric Waves through Tubes, 

 or the Vibrations of Dielectric Cylinders. By Lord 

 Eay leigh, F.R.S.* 



General Analytical Investigation, 



THE problem here proposed bears affinity to that of the 

 vibrations of a cylindrical solid treated by Pochham- 

 merf and others, but when the bounding conductor if 



* Communicated by the Author, 

 t Orelle, vol. xxxi.'l876. 



Phil. Mag. S. 5. Vol. 43. No. 261. Feb. 1897. L 



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