Theory  of  PJiasemeters.  97 
the  moving  coils  need  be  used  if  it  is  connected  to  its  appro- 
priate voltage.  The  deflexion  will  be  the  same  whichever 
moving  coil  is  connected  to  the  circuit.  For  unbalanced 
loads  it  is,  however,  necessary  to  have  all  the  coils  so  con- 
nected if  the  instrument  is  to  read  cos  <jy  as  defined  in  (9). 
To  calibrate  such  an  instrument  by  the  direct-current 
method,  the  process  already  described  may  be  applied,  the 
three  coils  o£  one  system,  but  only  one  coil  from  the  second, 
being  used  for  the  test. 
To  confirm  this,  tests  were  made  on  a  commercial  instru- 
ment of  the  symmetrical  six-circuit  type,  manufactured  by 
Messrs.  Everett,  Edgcumbe  &  Co.  The  moving  system 
consisted  of  three  star-connected  coils  designed  for  shunt 
connexion  with  the  mains.  A  constant  current  was  passed 
through  one  of  the  three  fixed  coils,  various  small  measured 
currents  were  passed  through  the  moving  coils  in  accordance 
with  the  method  above  described,  and  the  value  of  (p  corre- 
sponding with  each  observed  deflexion  x  was  calculated  by 
means  of  formula  (8).  The  scale  of  the  instrument  was  so 
constructed  that  with  alternating  currents  the  deflexion  x  was- 
a  direct  measure  of  the  phase  angle.  The  observed  value  of  x 
was  found  to  agree  with  cf>  when  a  particular  fixed  coil  was 
used  for  the  current  circuit.  Two  additional  sets  of  tests. 
were  taken  using  the  remaining  fixed  coils  in  succession.  In 
one  of  these  cases  </>  had  to  be  increased  by  120  degrees,  and 
in  the  other  diminished  by  the  same  amount,  to  make  the 
converted  value  of  <f>  agree  with  x.  The  tests  made  were 
comprehensive,  each  set  consisting  of  about  35  determinations 
ranging  over  the  whole  scale.  The  three  curves  obtained  by 
plotting  the  (converted)  value  of  cf>  with  x  were  nearly 
straight  lines,  and  were  nearly  the  same  ;  but  they  were  by 
no  means  absolutely  so,  and  the  differences  amounted  in  some 
cases  to  as  much  as  ten  degrees.  The  arrangement  of  the 
coils  was  thus  not  symmetrical  in  the  mathematical  sense 
assumed  in  the  above  proof,  and  it  is  difficult  to  estimate  to 
what  extent  the  indications  of  the  instrument  are  dependent 
upon  the  balance,  or  lack  of  balance,  of  the  load  currents. 
To  test  such  an  instrument  thoroughly  with  alternating  cur- 
rents on  unbalanced  loads  would  be  so  complicated  and 
laborious  that  it  is  unlikely  that  such  a  test  ever  has,  or  will 
be,  made.  There  would  be  needed  no  fewer  than  nine  elec- 
trical instruments  (three  ammeters,  three  voltmeters,  and 
three  wattmeters)  besides  a  number  of  troublesome  adjust- 
ments and  subsequent  calculations.  The  labour  of  the  nine 
or  more  observers  would  be  of  no  avail  unless  all  the  readings 
for  each  test  were  taken  accurately  and  simultaneously. 
Phil.  Mag.  S.  6.  Vol.  11.  No.  61.  Jan.  1906.  H 
