330 Mr Pocklington, Electrical Oscillations in Wires. [Oct. 25, 



with that of the wave. It is noteworthy that the amplitude of 

 the induced vibration is independent of the thickness of the 

 wire. If the incident wave is not proceeding in a direction 

 perpendicular to the ring, the problem can still be solved by a 

 method similar to the above. The vibrations induced in the 

 ring will however not be confined to the fundamental, but will 

 include vibrations of all modes. 



8. Helix. We will now consider the case of vibrations pro- 

 pagated along an infinite wire wound into a uniform helix. Let 

 the equations of the axis of the wire be x = a cos </>, y = a sin <£, 

 z — acf) tan co. We shall assume \ = A e'0*. This assumption is 

 justified later. The value of the force tangential to the wire at 

 the point {vr, 6, z) is e l $ e times that at the point (-sr, 6,z — ad tan co). 

 Hence (2) gives 



= Ae ipt 



\ e^ e ! d(j)L/3e l ^U + a 2 jcos a ! d<f>a cos 0e**II o 



/0 = O J -oo ( J — oo 



I d(f> a tan co e L ^Ii \ , 



+ sinw 



where II is the value that II takes when t = and cf> is put for 

 <j>-0. 



Hence = I dcpe 1 ^ {o?o? tan 2 » - B 2 + a 2 a 2 cos <f>) n o . . .(7). 



J — 00 



In obtaining this equation small quantities only have been neg- 

 lected. If however e/a is very small, we may in this equation 

 neglect all finite quantities in comparison with those of the order of 

 log e/a. In this case we may with advantage find an approximate 

 value of the right-hand side of (7). 



Assuming k any finite quantity, and neglecting terms that are 

 finite, the right-hand side of (7) is 



/: 



' 7i fi* e- Ma (t> tan co . , , 



aflbe'P* r-, era 2 tan 2 eo — B 2 + oca- cos q>\ 



o — a</> tan co l 



o?a? sec 2 to — /3 2 



V2a (a + e) (1 — cos </>) + a 2 </> 2 tan 2 eo + e 2 



r d<i> 



J —K 



r x gLaa4> tan w 



+ I d<be 1 ^ —r-, \cPa 2 tan 2 <w — B 2 + a?a? cos 6). 



J K T ' a<p tan co l 



The second integral is, neglecting finite quantities, 



— 2 (a 2 a 2 sec 2 w — B 2 ) log e. 



