1897.] Mr Pocklingtuii, Electrical Oscillations in Wires. 331 



The first and last can be reduced to sums of integrals of the 

 form I dcf> —7— . This integral * is — log 7 to our order, and thus 

 the first and last integrals give 



\(oi 2 a? tan 2 &) — B-) loa; (a 2 a 2 tan 2 co — 6 2 ) 



a tan co ° 



+ £a 2 a 2 log {(1 + /3 + aa tan co) (1 — /8 + aa tan to) (1 + /? — aa tan <w) 



(1 — /3 — aa tan co)}], 



and therefore the approximate form of (7) is 



2 (a 2 a 2 sec 2 w — /3 2 ) sin co log e = (a 2 a 2 tan 2 &) — /3 2 ) log (a 2 a 2 tan 2 &) — /3 2 ) 



+ |a 2 a 2 log {(1 + /3 + aa tan &>) (1 — /3 + aa tan co) (1 + /3 — aa tan co) 



(l-/8-aa tan «)}... (8). 



Several cases may occur, (i) In general, if a is not small, the 

 only term of importance is that on the left, so that 



a 2 a 2 sec 2 co — /3 2 = 0, or /3 = aa sec <o, 



and the velocity of propagation measured along the wire is 



-~ a sec w = — = V , 

 p a 



the same result as that obtained in the case of a circle. 



(ii) If however a and /3 are small, the first term on the right 

 is also of importance. If a and (3 are so small that log e can be 

 neglected in comparison with log (a 2 a 2 tan 2 co — /3 2 ), i.e. if the pro- 

 duct of the wave-length of the disturbance into the radius of the 

 wire is very large compared with the square of the radius of the 

 helix, we have a 2 a 2 tan 2 co — /3 2 = 0, or /3 = aa tan co, and the velocity 

 of propagation measured along the wire is 



7s a sec to = — cosec co = V cosec co, 

 p a 



so that the disturbance is propagated with a velocity V measured 

 along the axis of the helix. 



If a and f3 are small, but not so small that loge can be 

 neglected, i.e. if the product of the wave-length into the radius 

 of the wire is comparable with the radius of the helix, the 

 velocity of propagation has an intermediate value -f*. It is easy 

 to see that a like result does not follow if we try to make 

 (ora 2 tan 2 ft> — /3 2 ) small without making a and /3 small. 



* J. W. L. Glaisher, Phil. Trans., 1870, p. 369. 



t Hertz, Wied. Ann. 1889, vol. 3G, p. 21; Electrical Waves (tr. Jones), p. 158, has 

 proved experimentally that this is the case. 



