270 Mr. F. T. Trouton on the Acceleration 



The electric component of the wave* 



a f . , N cos (mr — nt) sin (mr—nt)~\ 



z= - < —BmOmr — nt) * - + ~-z — - r 



r [_ ' mr mrr* J 



may in like manner be considered. Similarly to before, if 

 2=0, it is necessary everywhere along the line considered that 



, . mr 



tan imr — nt) — - 5-^ 



y 1 — mr 



From this we get for the velocity, 



_ v m 4 r 4 — mV 2 + 1 



mV(mV- 2) 



In this case v has an infinite value both at the centre and at 

 distance X/7r^2 — roughly X/4'4. At intermediate points it 

 is negative or towards the centre. As the wave passes out- 

 wards it is at this point X/4'4 that the zero value of z is first 

 reached (at about '12 r before it becomes zero at the centre, 

 which occurs at t/2), whence it spreads both outwards and 

 inwards. On the diagrams this appearance of the zero value 

 of z is easily traced, and its subsequent outward movement at 

 the centre of the small circles ; these of course are the cross 

 sections of a vortex-ring, as it were, thrown off by the 

 " vibrator." 



For determining the time to reach any point, we have 



I — = ^ J r tan- 1 ^ 3-9 +const. > . 



J v V [ m 1 — nrr J 



This between NX and X/4'4, when N is a large number, gives 

 as the time taken, 



r(N- 1/4-4) -t/2tt . tan" V 2, 

 the first part being the normal time. 



If we wish from this to determine the period elapsing 

 between the arrival of the wave in this phase at the point ]SIX 

 and the time of zero value of the electric force at the centre 

 which occurs at the epoch of fig. 1, Plate VI., we must add 

 the epoch of the appearance at X/4'4. This will be found to 

 afford NT, the normal time. But if we reckon from the time 

 the zero value, having appeared at X/4*4, travelling inwards 

 reaches the centre, we obtain NT— t/2, an acceleration of one 

 half- period. In neither case, it is to be observed, does it 

 represent the time of passage between two places. The zero 

 value of z at the centre in fig. 1 ought really to be referred 



* Wiedemann's Annalen for 1889. 





