82  Dr.  W.  E.  Sumpner  on  the 
instrument  is  more  complicated  to  make  and  to  connect  to 
the  circuit,  but  it  possesses  certain  advantages  as  will  be 
shown  later.  With  rare  exceptions  the  magnetic  circuits 
of  phasemeters  are  air  circuits,  containing  no  iron,  so  that 
the  magnetic  "fields  associated  with  them  are  weak,  and  the 
instruments  in  consequence  are  of  somewhat  delicate  con- 
struction. There,  however,  appears  to  be  no  reason  why  the 
use  of  iron  should  be  avoided  in  these  instruments,  and  in 
what  follows  the  writer  proposes  to  show,  that  the  theory  of 
these  instruments  is  the  same  whether  they  contain  iron  or 
not,  and  however  the  coils  may  be  arranged  ;  that  they  can 
be  calibrated  by  direct  current  methods  although  for  use  on 
alternating  current  circuits  ;  and  that  a  new  type  of  instru- 
ment, containing  iron,  conforms  to  the  theory  given.  An 
investigation  will  also  be  made  of  their  action  when  the 
circuit  currents  are  of  unequal  magnitude. 
The  study  of  these  instruments  really  resolves  itself  into 
an  examination  of  the  behaviour  of  multiple  magnetic  circuits 
when  actuated  by  independent  currents. 
Theory  of  Four- Circuit  Phasemeters. 
We  shall  first  consider  the  case  of  a  three-phase  instrument 
having  three  fixed  coils  for  the  circuit  currents,  and  one 
movable  coil,  in  series  with  a  non-inductive  resistance, 
traversed  by  a  current  produced  by  one  of  the  circuit 
voltages. 
In  all  cases,  it  will  be  assumed  (i.)  that  the  induction 
density  at  any  point  due  to  the  current  A  in  a  fixed  coil  is 
represented  by  AF,  where  A  is  the  instantaneous  value  of 
the  current  A,  and  F  is  a  quantity  dependent  merely  on  the 
coil  and  the  position  of  the  point  considered  ;  and  (ii.)  that 
the  principle  of  superposition  holds,  viz.  : 
B  =  AiFj  +  A2F2  +  A3F3, 
where  A1?  A2,  A3  are  the  currents  in  the  three  fixed  coils  and 
the  quantities  F  are  functions  of  the  position  of  the  point 
corresponding  to  these  fixed  coils.  The  addition  must  be 
made  vectorially  if  the  subsidiary  fluxes  are  differently 
directed. 
There  is  of  course  no  doubt  about  either  of  these  assump- 
tions if  the  magnetic  circuits  associated  with  the  fixed  coils 
pass  wholly  through  non-magnetic  and  non-conducting  media 
such  as  air.  When  the  path  of  the  lines  of  force  lies  partly 
through  (suitably  laminated)  iron  it  is  necessary,  however, 
to  consider  the  effects  of  varying  permeability  and  hysteresis. 
But  any  magnetic  circuit  with  which   the  moving  coil  is 
