Theory  of  Phasemeters.  103 
Bat  from  the  above  we  have 
60  =  e2i  cos/30, 
or  0O- -62*008(20-/3^ 
or  0O=  — [0t  cos20  +  e1  sin  20], 
'here 
01  e2-e3  €2  — e3 
and  the  fractional  error  made  in  reading  cos  0,  which  by  (24) 
is  —  00  tan  0,  becomes 
A  C0S/  =  e2*  cos  (20-A)  tan  0,   .     .     .     (32) 
COS0  \      T        I      /  T1 
in  which  the  quantities  e  and  /3j  are  determined  solely  by  the 
divergences  of  the  load-currents  A1?  A2,  A3  from  their  arith- 
metical mean  value. 
The  same  formula  can  be  obtained  by  equating  any  two  of 
the  ratios  (22)  without  bringing  the  quantities  A0  and  e0  into 
the  equations,  but  the  working  is  not  any  shorter  and  the 
information  yielded  is  less. 
It  will  be  apparent,  on  inspection  of  the  error  formula  (32), 
that  the  phasemeter  may  give  very  erroneous  readings  when 
the  load-currents  are  badly  out  of  balance.  This  can  be  seen 
best  by  considering  a  numerical  case.  Suppose  the  three 
load-currents  are  21,  22,  and  17  amperes.  The  mean  current 
is  20,  and  the  errors  el9  e2,  e3  from  the  mean  are  5,  10,  and 
15  per  cent,  respectively.  It  follows  that  e  is  10*8  per  cent. 
and  e2*  is  15*3  per  cent.  Disregarding  for  the  moment 
the  factor  cos  (20  — Z^),  the  above  error  has  to  be  multiplied 
by  tan  0  to  get  the  percentage  error  in  reading  cos  0.  The 
error  will  thus  be  much  reduced  on  circuits  of  high-power 
factor,  but  for  power-factors  below  0*71  the  value  of  tan0 
becomes  greater  than  unity,  and  the  percentage  error  in 
reading  cos0  will  be  correspondingly  increased,  though  it 
must  be  noted  that  if  the  scale  is  graduated  to  read  cos  0  the 
absolute  error  of  the  reading  in  the  case  assumed  is  never 
greater  than  0*153  sin0. 
Now,  a  circuit  having  three  currents  proportional  to  those 
assumed  would  be  considered  badly  out  of  balance,  and  as  a 
rule  a  much  better  state  of  things  obtains,  yet  the  conditions 
instanced  are  quite  possible  in  practice.  The  influence  of 
the  factor  (cos  20—^)  must  also  be  considered.  This  factor, 
though  it  must  always  decrease  the  magnitude  of  the  error, 
can  alter  its  value  in  a  striking  manner,  and  is  the  quantity 
