Potentials,  Molotropic  and  Isotropic.  573 
And  since 
dK  AW  =  A  M  =    a7  '  and  "55/ A W  =  2A  ^  =  "" a7^ 
we  have  r\    Ar  \  r  \      W*A(^) 
Also  ?<A(^)  written  at  length  is 
Xa*{  (BO-  A'2)  +  K<?C  +  7-B  -  2p'AQ  +  /*2P} 
+  2%r  {B'C  -  AA'  +  /^'C  +  r'W  -pA'-p'A)  +  ^F}, 
and  therefore 
~  ua^j  -  (C  +  iir)f  +  (B  +  /*?)  .:*  -  2  (A'  +  /^'fc  j 
1    d  <6> 
2^a'  MA(/i)  =  (C  +  yur')^  +  (B;  +  ^)ys  -  (A  +  ^)ys -  (A'  +  fip')x2. 
When  the  whole  operator  is  used  all  terms  are  cancelled, 
and  we  have  the  three  results 
AW 
A., 
0  .  uA(l,m,n)  =  {Ix  +  my  +  nzf,  0  .  Afr)  =  UA(l,)(x,y,  z) 
and  O.uA(fl-)(:x,y9z)==0. 
Hence  the  operation  applied  to  the  first  and  third  members 
of  (4)  yields  a  repetition  of  (4)  with  s-f-1  for  5;  and  (4) 
follows  by  repeated  use  of  the  operator  starting  from  (5). 
The  formula  of  transference  when  ax  4-  y'y  -f  ft'z  appears 
under  the  integral  sign  is  got  from  (4)  by  use  of  the  operator 
r\  i~      d       y    d        z    d 
Ux  =  *dA  +  2dU+2  dB" 
for  which 
Ox'.uA=l(lx  +  m.y  +  nz),    0/  .  A  fcc)  =^  {*x  +  </t,  +  ffz)9  1    (g) 
OJ  .uA(fJi)  =  0.  J 
This  gives 
_1_  C  l(l3c±7ny  +  nz)2s-1dco  _  u'1  (ax  +  y'y  +  fiz) 
47rJ    '      (uA+fiup)s+W        -  (2s  +  l)AWA(j*)  '      () 
Integrate  with  regard  to  \x  from  /j,  to  oo  and  raise  s  by  1, 
then 
(.7) 
(10) 
