278 
Transactions of the Royal Society of South Africa. 
occur, the last exponential curve must not rise above V,„„^ (fig. 3). We have 
V. = (an + y)V, 
V,« = (am + yWe (y = displacement of abscisse axis) 
aY = 2dYe 
Place V, = 1, then V„ =: a„ + 7 ; V„, = + y, AY = 28. 
We found previously 
te»0 = K = ^=^| 
(an — Om — ^)I 1 
= ^, whence 
t' = (Cln - 0,n - S) (T + . . . (11) 
T is the time taken from a = 0 to a = 1, /. e. for the whole range of 
regulation ; hence in t' seconds the value of I varies by ^ x I amperes, 
T -j- t° 
or, according to equation 11, l)y - (a„ — a^^ — f^)I amperes. 
In time t' the factor a has also increased by the amount 
The value at the beginning of the regulating period was a,„, hence the 
factor of the last exponential curve, which cuts the line Y„n)i. in C, is 
am + "^-^^-^ i^n — am — 8). 
If now the maximum value of the pressure is not to exceed Y.nax., then 
we must have — 
T -\- t" = 
ctm -f (an — ctm " S) ^ a„ 4 wheuce ' 
T - ^^(«^ - i ), and inserting the value ^ for t,, 
T^igr^"-^" - 1) (12) 
Equation 12 holds for rising currents. The curves for falling values 
are on the whole flatter than the rising ones, and excess regulation and 
hunting need not be feared if it does not take place for rising currents, 
especially as load increases are mostly in excess of load decreases. But it 
is important that no new load rush occurs before the previous one has been 
dealt with, the danger of hunting being especially great when a load drop 
follows rapidly on a load rise. The interval t' follows from equation 11, 
viz. : 
t' > (a, - am - a) (T -f tj, 
and inserting the value of T from equation 12, we obtain — 
i 
