84  Dr.  TV.  E.  Sumpner  on  the 
magnitudes  o£  the  currents  A,  and  which  must  also  he 
unaffected  by  the  magnitude  of  V.  For  alternating  currents 
the  position  of  balance  can  only  depend  on  the  phase  dif- 
ferences of  the  currents  A  with  regard  to  the  voltage  V, 
and  on  the  relative  magnitudes  of  these  load  currents.  For 
direct  currents  the  equilibrium  value  of  x  can  only  depend 
on  the  ratios  between  A1?  A2,  and  A3. 
Now  in  the  theory  of  phasemeters  as  hitherto  given,  the 
following  assumptions  are  made  : — 
(i)  That  the  currents  and  voltages  vary  with  time  accord- 
ing to  the  sine  law. 
(ii)  That  the  functions  F  are  of  equal  magnitude  and  vary 
with  the  deflexion  x  according  to  the  sine  law. 
(iii)   That  the  load  currents  are  balanced  between  the  phases, 
that  is  to  say,  the  magnitudes  of  the  currents  A1? 
A2,  and  A3  are  equal. 
It  is  well  known  that  assumptions  (i)  and  (iii)   are  only 
true  in  practice  under  most  exceptional  circumstances,  while 
assumption  (ii)  is  only  true  when  the  moving  coil  is  of  very 
small  dimensions   compared  with  those  of  the  fixed  coils  and 
is  placed  at  their  common  centre.     This  cannot  be  the  case 
in  an  actual  instrument.     In  what  follows,  we  shall  not  find 
it  necessary  to  make  any  of  these  assumptions,  but  we  shall 
confine  ourselves  at  present  to  the  case  of  balanced  loads  for 
which  the  magnitudes  of  Alf  A2,  and  A3  are  equal. 
By  the  magnitude  A,  of  any  cyclic  function  A,  of  a  variable  t 
and  of  period  P,  we  understand  that  quantity  whose  square 
is  equal  to 
A2  =  ~C \2dt. 
P  o 
It  follows  from  the  foregoing  that  with  balanced  loads  the 
condition  of  equilibrium  at  the  deflexion  x  is 
F1A^V  +  F2A2V  +  F3A3V=0,   ....     (3) 
and  is  such  that  the  coefficients  of  F1?  F2,  F3  are  proportional 
to  quantities  which  merely  depend  on  the  phase  differences 
between  the  currents  and  the  voltage  V,  and  also  that  if  we 
pass  steady  currents  through  all  the  coils  and  adjust  those 
through  the  fixed  coils  till  they  are  proportional  respectively 
to  A:V,  A2V,  A3V,  we  shall  get  the  same  deflexion  as.  A 
given  deflexion  thus  always  corresponds  with  a  particular 
power  factor  of  the  load  ;  and  in  order  to  calibrate  the  instru- 
ment by  means  of  direct-current  tests,  it  remains  to  show 
how  to  calculate  the  power  factor  from  the  ratios  of  the 
direct  currents  used  in  order  to  produce  the  corresponding 
deflexion  of  the  instrument. 
