Discharges in Electrodeless Vacuum-Tabes. 517 



with larger jars. Now the E.M.F, acting in the rarefied gas, 



and producing the breakdown of the same, is proportional to 



dc 



dt' 



dc 

 Also the greatest value of -j- that first occurs is when £ = 0, 



and then 



tte_V 

 dt " L> 



and the next is for very rapid oscillations 



dc V _?* / 



dt=Z e 2V 



So that the first value of the E.M.F. acting in the gas is 

 independent of the capacity, and the next and succeeding 

 values are less the greater the capacity. 



The effect on the eye, however, of the luminous ring will 

 be the time -integral of the discharge or approximately depend 

 on 



dc , 



di dt - 



i 



The whole limits of t, viz., from to go , cannot be taken at 



dc 

 once, as -7- keeps reversing, and this reversal will not affect 



the luminous discharge. Referring to the curve (fig. 9) it 

 will be seen that the first reversal must take place at Mj, when 

 bt = 6, and subsequent ones for values 7r + 6, 27T + 0, &c.,of bt. 

 It is therefore necessary to take first the limits 6 and 0, then 

 it + 6 and 6 ; 2ir-\-0 and ir + 6, and soon, alternately writing 

 the integrals plus and minus. 



f °° do V f e 7r + r "1 



J. §^ = 5i{ ^ sin ^- lf ts[abt l 



+ e at sin bt — &c. ad inf. > t 



Remembering that 



sin ( 7r + 0) = — sin #, 

 sin(27r + 0)= sin0, 

 sin (o7r-f 0) = —sin 0, and so on, 



