110 
MESSRS. J. HOPKINSON AND E. WILSON OX THE 
cliarge and cajjacity. Suppose that from ^ = 0 to t — t, Xi = K, and before that 
time Xi — 0, then i/i = ^ \ \p (w) doj, and ' ‘ = xjj (^t) ; thus ifj (t) is the conductivity 
after electrification for time t. It has of course been lone- known that in statins; the 
conductivity or resistance of the di-electric of a cable, it is necessary to state the 
time during which it has been electrified ; hence x(/ (t) is for many insulators not 
constant. ^ may perhaps be defined to be the true conductivity of the condenser, 
but at all events we have ijj (t) as the expression of the reciprocal of a resistance 
measurable, if we please, in the reciprocal of ohms. For convenience we now 
separate i/j (oo ) = /5 from ip (w) and wrii;e for ip (oj), xp (oj) + If we were asked to 
define the capacity of our condenser we should probably say : “ suppose the con¬ 
denser be charged to potential X for a considerable time and then be short-circuited, 
let Y be the total quantity of electricity which comes out of it, then Y/X is the 
capacity.” If T be the time of charging = X [xp (w ) + ^} do) at the moment 
j f\ 
ri +1 
of short circuitiniLr ; = X ) + /3; doj after time t of discharge. The 
"is ^ 
* t 
amount which comes out of the condenser is the difference of these, or 
Y=x|| xp [(x})-\-^do} — xp (oj) fi- /I dojj ; if i be infinite xp (^) = 0 , and Y=X [ xp (oj) dw ; 
or we now have capacity expressed as an integral of ^ (co) and measurable in micro¬ 
farads, and it appears that the capacity is a function of the time of charge increasing 
as the time increases. Experiments have been made for testing this point in 
the case of light flint glass, showing that the capacity Avas the same for 1/20000 
second and for ordinary durations of time (‘ Phil. Trans.,’ 1881, p. 356), doubtless 
^ _ y/ 20 oco rt 
xp(co)do} is small compared with xp(ct))dco. Noav xp (oj) do, when t 
1/20000 Jo Jo 
is Indefinitely diminished, may be zero, have a finite value, or be infinite ; in fact it 
has a finite value. The value of xp (o) when o is exti'emely small can hardly be 
observed; but xp (o) do, when t is small, can be observed. It is therefore coii- 
venient to treat that part of the expression separately, even though we may 
conceive it to be quite continuous with the other parts of the expression, f xp (o) do, 
• 0 
when t is less than the shortest time at which we can make observations of xp (oj), 
is the instantaneous capacity of the condenser. Call it K and suppose the form of 
xp to be so modified that for all observed times it has the observed values, but so 
that \ ^ { o) do = 0 , when t is small enough. 
J 0 
Then i/i = Ka-j Xi_^ [x/j (o) -f- /3} do. Here the first term represents capacity, 
J 0 
the second residual charge, the third coiiductlvity, separated for convenience, though 
