286 Trmisadions of the Eoyal Society of South Africa. 
of ia. The number of vibrations does not matter, although to each value 
of ia corresponds a definite number of vibrations, found as follows. 
From equation 14 we obtain : 
dia a^tn 
Similarly from 15 
dt. 
di'a- 
la 
But dia 
dia and ia 
dt. 
dt. 
dia 
The reciprocal of this is 
dia 
i'a, whence 
\uj. — ia 
acyl 
^ (a^I — ia){la - « J) ^ ^^.^^ . 
dt^ + dto fol*^! — ^2)^« " 
= f is the number of vibrations of the system. 
(17) 
dt^ + dt.. 
To each value of ia corresponds a definite frequency /, but to a given 
frequency / correspond two values of ia. From equation 17 follows that 
f —O for ajl — ia and aol = It is a maximum for dia — constant and 
ia = IVa^ao. 
Example. — The generator in the previous example is to be regulated with 
a Tirrill instrument and the voltage is to be kept constant at 650 volts, 
within the limits of ± | per cent. In this case dia corresponds to 5*5 volts. 
For 550 volts the exciting current is 8 2 amperes. When resistance is in 
circuit the voltage is 50 per cent, of the normal, or 275 volts, and when it is 
out of circuit it is 118 per cent., or 650 volts. The corresponding exciting 
currents are 2 '8 and 14 amperes respectively. We have: 
= 1 ; rxo = 0'2 ; ia — 8*2 amperes, I = 14 amperes. 
The results are tabulated. 
Table. 
Time constant to 
3-68 
3-63 
3-55 
3-45 
3-15 
2-86 
2-55 
2-40 
io. 
2-8 
5-0 
70 
8-2 
9 
10-5 
12-0 
14-0 
amperes. 
(It I 
dt^ 
0-000 
1-222 
3 00 
4-65 
6-20 
110 
23-0 
00 
dia 
dt^ + dl^ 
0-000 
1-36 
1-50 
1-38 
1-37 
1-12 
0-75 
0-000 
The value of dia near 8-2 amperes is 0-213, hence the frequency of 
1'38 
vibration is ^..-^-j^g = 6*5 per second. 
