100  Dr.  "W.  E.  Sumpner  on  the 
unbalanced  load  assumed,  be  cos  6  as  determined  from  (22), 
and  that  this  value  is  absolutely  independent  of  the  quantities  F 
resulting  from  a  particular  structure  of  the  instrument.  In 
other  words,  if  a  number  of  such  phasemeters  be  connected 
to  the  mains  in  precisely  the  same  manner,  and  if  these 
instruments  have  all  been  calibrated  for  balanced  loads,  each 
of  the  instruments  will  indicate  the  same  reading  whatever 
the  nature  of  the  load,  and  however  different  the  various 
instruments  may  be  as  regards  internal  structure.  But  it 
does  not  follow  that  this  common  reading  correctly  gives  the 
power-factor  of  the  load. 
There  are  certain  relations  between  Alf  A2,  A3,  fa,  <f>2,  fa, 
resulting  from  the  geometrical  properties  of  fig.  6,  but  it 
cannot  in  general  be  true  that  the  value  of  cos  6  determined 
from  (22)  is  the  same  as  that  of  cos  0  as  defined  in  (9). 
This  can  readily  be  seen  by  putting  A3  =  0,  in  which  case  0 
must  be  30  degrees  while  cf)  may  have  any  value.  It  may 
all  the  same  be  the  fact  for  small  divergences  of  the  load 
currents  from  A  their  mean  value,  that  cos  0  and  cos  </>  differ 
but  very  slightly.     We  thus  want  to  find  0O  where 
0  =  <£  +  0O, 
since  if  0O  is  small,  the  error,  considered  as  a  fraction  of  the 
true  power-factor,  made  in  taking  cos  6  (the  reading)  to 
represent  cos  </>  (the  true  power-factor)  \\  ill  be  : 
r   -—rr    =  —  #o  tan  fa    ....     (24) 
cos^       dcj)  T  v     y 
and  0O  wiH  correspond  with  the  phase  error  of  an  ordinary 
wattmeter  due  to  inductance  in  its  pressure-coil.  For  equal 
phase  errors  the  two  instruments  will  read  erroneously  by 
precisely  the  same  percentage  on  loads  of  the  same  power- 
factor.  The  questiou  to  determine  is  whether  0O,  for  a 
moderately  unbalanced  load,  is  sufficiently  great  to  render  the 
phasemeter  unsatisfactory. 
While  0O  is  small  the  instrument  remains  a  phasemeter, 
but  when  60  becomes  large  the  instrument  tends  to  indicate 
the  want  of  balance  of  the  currents,  rather  than  the  average 
power-factor  of  the  load. 
If  we  put  each  of  the  ratios  in  (22)  equal  to  A0  and  con- 
sider fig.  6,  it  will  be  noticed  that  A0  is  the  length  of  each  of 
the  sides  of  an  equilateral  triangle  drawn  so  that  these  sides 
make  angles  6  with  the  sides  of  the  triangle  representing  the 
voltages,  and  so  that  the  vertices  lie  on  lines  perpendicular 
to  the  voltage  V1?  and  passing  through  the  angular  points  of 
the  current   triangle.      For   balanced   loads   the  A0  triangle 
