104 
Dr.  W.  E.  Sumpner  on  (Ji< 
which  determiDes  whether  the  instrument  reads  high  or  low. 
The  quantity  e  in  (32)  depends  merely  on  the  average  square 
of   the  fractional  divergences   € 
i,  e2,  e3  of  the   load-currents 
A1?  A2,  A3  from  their  mean  value.  The  quantity  ft,  on  the 
other  hand,  does  not  depend  upon  e  but  upon  the  ratios 
between  e1?*  e2,  e3.  Whatever  the  value  of  e,  ft  may  have  any 
value  in  accordance  with  these  ratios,  so  that  cos  (2(f)  —  ft) 
varies  sinuously  with  ft  and  will  be  numerically  equal  to 
unity  for  a  particular  relation  between  ft  and  </>.  Even  for 
the  same  three  currents  in  the  mains,  ft  may  have  no  less 
than  six  values,  since  we  can  select  any  of  the  three  currents 
for  the  line  A1?  and  after  this  selection  is  made  we  may 
choose  either  of  the  two  remaining  currents  for  the  line  A2. 
The  six  possible  values  of  ft  for  the  currents  assumed  in  the 
above  case  can  be  shown  to  be  +19°,  ±41°,  and  +79°,  and 
the  percentage  errors  for  these  values  of  ft  and  for  a  value 
of  e  equal  to  10  per  cent,  are  shown  in  Table  III.  for  loads 
of  different  power-factor.  Another  column  is  added  for  the 
case  ft  =  0,  and  in  the  last  column  the  maximum  error 
e.2i  tan  <£>  is  shown. 
Table  III. — Percentage  Error  of  Phasemeters 
(for 
a  load  10  pe 
r  cent. 
out  of  balance) . 
Power- 
factor. 
Values  of  f3l  in  degrees. 
Maxi- 
mum. 
-79. 
-41. 
-19. 
0. 
+19 
+41. 
+  79. 
1-0  ... 
0 
0 
0 
0 
0 
0 
0 
0 
•9  ... 
6-1 
6-7 
5-7 
4-2 
2-2 
-  -3 
-4-4 
69 
•8  ... 
10-6 
8-9 
61 
30 
-  -5 
-45 
-9-4 
10  6 
•7  ... 
14-1 
9-2 
4-3 
-  -3 
-5-0 
-9-6 
-14-2 
14-4 
•6  ... 
169 
8-0 
•8 
-5-3 
-10-9 
-16-0 
-18-9 
189 
•5  ... 
18-4 
4-6 
-4-6 
-122 
-18-4 
-23-1 
-23-6 
24-5 
It  will  be  seen  that  the  error  may  be  quite  serious.  Thus 
a  14  per  cent,  error  on  a  load  of  power-factor  0'7  means  that 
the  instrument  may  read  either  0*8  or  0'6  according  as  the 
error  increases  or  decreases  the  reading.  Of  course  it  must 
be  remembered  that  the  error  is  always  proportional  to  e, 
which  may  be  called  the  fi  out-of-balance  "  of  the  load  ;  so 
that  if  this  out-of-balance  is  2  per  cent,  instead  of  10  per 
cent.,  all  the  above  numbers  must  be  divided  by  5.  In  all 
cases  the  reading  for  a  power-factor  cos  <p  fluctuates,  according 
