1887.] Electric Time-constant of a Circular Disk, 293 



Some further information may be gathered from the second 

 principle stated above. The electrokinetic energy of the system of 

 currents denned by (3) is — 



T = A ~ — . 27rrdr 

 2 J dr dr 



The integral 



r(^+m).r(|+m) r(2) [\ m+l(l _ z)n - idz 



-riw^i s r(-^+^)r(|+m) l_ 



^ " f " 2; • Ml r(m+i)r(-^)r(f) r(m+»+|) 



_ r(w +^) 



- r(w+j)r(2w+i) 

 r(2n+f)r(»+i)' 



Hence T = ^ . *^±1> (12.) 



If the disk be of uniform resistance //, the dissipation is — 



= T P • S ^T7 rr r s(l— *0 l dz 



»y ;r( w +i)\ 8 fl3) 



2w(2w + l) lro+^) J v 



Introducing a time-factor e~^ T , and supposing that the system of 

 currents is constrained to remain of the type (3) during the decay, 

 we find on equating the rate of diminution of the energy to the 

 dissipation — 



_ Trfa r(2n + 2){r(n+j)}8 Q4 x 



~ p' r(»)r(«+i)r(2»+*y v -J 



Any value of t obtained from this formula will be an inferior limit 



