The Theory of Aidomatic Regulators 
275 
but is displaced to the right by a distance or time cl^ (see fig. 2). Time is 
required before the current is switched in. The second curve follows 
later than the first one, being thus displaced from the origin by — '^d^. 
Obviously, these times will be the smaller the larger the number of contacts. 
On drawing these curves it will be found that all possess the same tangent, 
assuming our previous assumptions are fulfilled. Let this tangent form 
with the abscisse axis the angle 6, then 
tan 6^ = K = constant . . . • (^) 
where the value of K is found as follows : 
The equation of the exponential curve 2 is given by 
Fig. 2. — Regulation curves. 
dt / ' "'^ 
t ato — \ — ^ 
uL 
whence dy 
dx~ t 
(8) 
and as our curve is displaced to the right by a distance dd^ we have 
(placing ia = y and t = x — d) 
X — d \ 
y = al\l — e / 
whence _ I - ^ ~ ^ 
d 
From equation (8) follows 
