276 
Transactions of the Hoyal Society of South Africa. 
The equation of the common tangent which envelops the various 
exponential curves is 
y' =f{^') 
which has, in common with the exponential curves in the tangent points, 
the ordinates y' = y, the abscissae = or, and the differentials dx' = dx and 
dy' = dy. With our assumptions of uniform switching speed and spacing 
we must obtain uniform variations of the exciting current, and dx' must be 
proportional to da, hence x' rises from Oto T proportionally to a from Oto 1, 
so that x' — aT, and for any tangent point 
dx' to ' X I / 
Fig. 3. — Eeofulation curve. 
This equation is fulfilled by 
y' = 'K.x 
y' (^y' 1 
— = K- = — , whence 
dx 
_ ] 
(10) 
Consider now the process of regulation. Let the normal pressure, say at 
no load, be corresponding to a definite position of the regulator. Let 
this be given for n = 0.5 = /. e. the exciting current is equal to half the 
