Theory  of  Phase  meters.  85 
Now  however  the  instrument  may  be  calibrated,  its 
readings  will  only  be  correct  when  its  coils  are  all  connected 
to  the  circuit  in  the  particular  manner  corresponding  with 
the  calibration.  We  must  thus  adopt  for  each  fixed  coil  one 
direction  as  positive,  and,  i£  we  distinguish  the  ends  of  the 
coil  as  positive  and  negative  respectively,  we  shall  under- 
stand by  a  positive  current  through  the  coil  a  current  flowing 
from  its  positive  to  its  negative  end.  Moreover,  if  we 
consider  a  current  in  the  mains  as  positive  when  flowing  in 
a  particular  direction  along  the  mains,  we  have  the  following 
equation  true  at  every  instant,  whatever  the  law  of  variation 
of  the  alternating  currents  may  be  : 
A1  +  A2  +  A3=0 (4) 
From  this  it  follows  that  the  mean  products  forming  the 
coefficients  in  equation  (3)  must  be  connected  by  the  relation 
A^  +  A^r  +  A^=0.      .....     (5) 
That  is  to  say,  in  the  calibration  by  means  of  direct  currents 
we  must  use  three  steady  currents  through  the  fixed  coils 
such  that  their  algebraic  sum  is  always  zero.  This  can 
easily  be  arranged  by  connecting  together  all  the  negative 
ends  of  the  coils,  and  afterwards  putting  two  of  the  coils  in 
parallel  through  variable  resistances,  and  in  series  with  the 
third.  The  two  selected  for  parallel  connexion  may  of  course 
be  varied  if  necessary  to  calibrate  the  instrument  throughout 
the  scale. 
But  there  is  another  consequence  of  (4)  the  truth  of  which 
is  also  quite  independent  of  any  assumption  in  regard  to  the 
variation  of  the  currents  with  time.  Equation  (4)  can  be 
regarded  as  a  vector  equation  such  that  if  three  vectors  are 
drawn  forming  a  triangle  the  sides  of  which  are  respectively 
proportional  to  the  magnitudes  of  the  three  currents,  the 
angles  of  this  triangle  will  perfectly  represent  the  phase 
relations  of  these  three  currents.  It  is  also  possible  to  find 
another  vector  representing  the  voltage  V  in  both  magnitude 
and  phase  so  that  the  mean  product  of  any  two  of  the  quan- 
tities considered  is  accurately  equal  to  the  scalar  product  of 
the  corresponding  vectors  *.  In  all  ordinary  cases  the  vector 
V  can  be  regarded  as  in  the  same  plane  as  the  current 
vectors,  even  if  the  currents  vary  in  a  manner  widely  de- 
parting from  the  sine  law  ;  but,  in  any  case,  if  Vp  is  the 
perpendicular  projection  of  the  vector  V   on  the   plane   of 
*  See  "The  Vector  Properties  of  Alternating-  Currents  and  other 
Periodic  Quantities,"  Proceedings  Royal  Society,  1897,  vol.  lxi.  p.  455. 
