98  Dr.  W.  E.  Sumpner  on  the 
Symmetrical  Five-Circuit  Pliasemeter. 
Before  leaving  the  consideration  of  symmetrical  instruments, 
it  may  be  well  here  to  state  the  result  of  an  examination  of 
the  case  of  one  having  five  coils  instead  of  six. 
If  the  first  coil  of  the  moving  system  is  omitted,  making 
Fn  =  F21  =  F3i  =  0  in  equations  (10),  (11),  and  (12),  it 
will  be  found  on  making  the  changes  which  necessarily 
follow  that  equation  (13)  becomes 
P[O-A1cos^1]=q[C^3^-A3cos((/)3  +  60)] 
+  Rp^-A2cos(4>2-60)]; 
now  in  the  above  equation  the  coefficient  of  P  is  equal  to 
the  sum  of  the  coefficients  of  Q  and  R  (see  (21)  and  (22) 
below),  and  we  can  thus  always  find  an  angle  x  such  that 
P  cos  x  =  Q  cos  (60  +  x)  +  R  cos  (%-  60), 
and  hence  by  (IS)  the  reading  of  the  pliasemeter  will  be  that 
due  to  a  balanced  load  of  power-factor  cos  X- 
With  the  help  of  (17)  and  relations  proved  below  (211, 
(22),  and  (25),  it  will  be  found  that 
3 A  cos  cj>  -  A0  cos  6  _  3A  cos  (0  -f  60)  - A0  cos  (6  +  60) 
cos%  cos  (%+60) 
_  3  A  cos  (<ft-60)-A0cos  (fl-60) 
cos(%-60) 
and  assuming  that  6  and  x  exceed  </>  by  small  amounts  60  and 
Xq  respectively,  it  will  be  found  that  the  above  equations 
involve  the  relation 
Xo  =  iA       (19) 
or  the  phasemeter  on  an  unbalanced  load  will  read,  instead  of 
the  true  power-factor  cos  (f>,  the  value 
cos(</>  +  J0o), 
where  60  has  the  value  given  by  (31)  below. 
Phase  Error  due  to  Unbalanced  Loads. 
If  the  instrument  is  unsymmetrical  and  of  less  complicated 
construction  than  those  just  considered,  the  next  best  form 
will  be  one  having  a  three-circuit  system  for  the  currents, 
and  a  single-circuit  system  for  the  volts,  like  the  one  first 
described.     Referring  to  equation  (3)  and  assuming  that  the 
