Discharges in Elcctrodeless Vacuum- Tubes. 519 



Now x and y will in general be small fractions, since KR 2 

 is usually much less than 4L. 



If the oscillations are very rapid, 6 is very nearly equal to 



- Hence y = 2# approximately. Then tf — e*^ becomes 



2# + — =2^1+^jappi 



Therefore the time-integral 

 2V 



ox. 



Vl V /K/ ^ 



=_ = — a / _ I 1— — I approx., 



and 



t / KB 2 tR /K 



so time-integral 



4V A 



4 V / n tt'R'K \ 



Now from this it is seen that the effect of increasing the 

 capacity would be to slightly diminish the time-integral, and 

 consequently probably make the brilliancy of the luminous 

 discharge less, if it were not that increasing the capacity 

 diminishes the real resistance of the circuit, since it makes 

 the oscillations slower, and the resistance R for copper for 

 rapid oscillations approximately equals \Z^UR ; where I is 

 the length of the wire, and R its resistance for steady cur- 

 rents. Now b — ,— approximately. 



Therefore 



a=\A' 



2^KL 



so that the time-integral is very roughly proportional to the 

 fourth root of capacity. 



There is also another reason why larger jars might produce 

 a brighter discharge, even though the time-integral were less. 

 With larger jars the time taken for the amplitude of the cur- 

 rent to sink to a value at which it becomes insignificant will 



