Since  tan  0^  ^^,  and  tan  0^  =-?i^,

it  follows  by  an  easy  transformation  that

RS-/.^LN

To  find  the  value  of  <^,  suppose  that  whilst  E  and  Ij  remain
the  same,  the  secondary  circuit  is  suppressed.  We  should
then  only  have  an  impressed  electromotive  force,  E,  creating
a  current,  Ii,  and  from  the  diagram  and  from  what  has  been
before  explained  it  is  obvious  that  the  effective  and  self-induc-
tive electromotive  forces  in  the  circuit  would  then  be  repre-
sented by  0  Q'  and  Q'  P'.  If  we  denote  these  by  the  symbols
B'li  and  L'plj  we  may  properly  caU  R'  and  L'  the  equivalent
resistance  and  inductance ;  that  is  to  say,  these  quantities  are
the  resistance  and  inductance  which  the  primary  circuit  should
have  in  order  that,  when  there  is  no  secondary  circuit,
the  primary  impressed  electromotive  force  may  generate  in
it  the  same  current  which  it  does  when  the  secondary  circuit
is  present  and  the  primary  has  its  natural  resistance  R  and
inductance  L.  We  see,  then,  that  the  effect  of  bringing  up
the  secondary  and  allowing  it  to  be  acted  upon  and  react

n2


180  SIMPLE  PERIODIC  CURRENTS.
upon  the  primary  is  to  increase  the  effective  resistance  and
diminish  the  effective  inductance  of  the  primary ;  in  other
words,  the  equivalent  resistance  of  the  primary  circuit  i^
greater  and  the  equivalent  inductance  is  less  by  reason  of
the  presence  of  the  secondary  circuit.
We  have  then  to  find  the  value  of  cos  {<t>  +  Ot).
