9G  Dr.  W.  E.  Sampner  on  the 
we  have  for  the  different  angles  required : — 
/\  /\  /\ 
^  =  0^    VxA2  =  02  +  12O,    V1A8  =  03+24O,     &c,     &c. 
If^  we    substitute    in    the   above    equations,    using   such 
relations  as 
/\ 
V1A3  =  VA3cosV1A3, 
and  divide  out  by  V  we  get, 
PX Ai  cos  </>!  +  QX A3  cos  (03  +  240)  4-  RSA2  cos  (<f>2  + 120)  =  0") 
or  > 
PS Ax  cos  0!  =  Q2A!  cos  (0X  +  60)  +  R2 A2  cos  (60  -  0X)     ) 
Let  us  define  two  quantities  C  and  S  such  that 
C  =  Ai  cos  0i  +  A2  cos  02  +  A3  cos 03| 
S  =  Ax  sin  0!  +  A2  sin  02  +  A3  sin  03  j 
We  then  find  on  expanding  (13)  that 
PC  =  QC^RC±p;    .    .    .    (15) 
but  from  (14)  we  get 
C2  +  S2  =  A12  +  A22  +  A32  +  2XA2A3cos(02-03)    .     (16) 
Now  we  only  propose  to  consider  the  case  of  a  moderate 
want  of  balance  for  which  the  angles  01?  02,  03  differ  from 
their  mean  value  by  only  small  amounts  whose  squares  and 
products  can  be  neglected  compared  with  unity.  Under 
these  conditions,  the  cosine  of  the  difference  of  any  two  of 
these  angles  can  be  considered  unity,  and  (16)  reduces  to: — 
C2  +  S2=;(A1  +  A2  +  A3)2, 
but  from  (9)  and  (14) 
C  =  (A1  +  A2  +  A3)cos0) 
S  =  (A1  +  A2  +  A3)sin0j 
If  now  we  substitute  in  (15)  and  simplify  we  obtain  : 
Pcos0  =  Qcos(6O  +  0)+Rcos(6O-0),       .     (18) 
and  substituting  from  (11) 
P  =  Fn  ,     Q  =  F3i ,    R  =  F2i , 
we  have 
Fu  cos0  =  F21  cos  (60-0) +  F31  cos  (60  +  0)  ; 
an  equation  exactly  similar  to  (6)  and  showing  that  the 
deflexion  is  the  same  as  on  a  balanced  load  of  power-factor 
cos  0.     Also  it  appears  that  for  balanced  loads  only  one  of 
