Vibrations  of  Conducting  Surfaces  of  Revolution.        705 
In  order  that  these  equations  should  be  mutually  compatible, 
the  solenoidal  conditions 
A     JL  +  A     -^—=0 
~ba' PzPz      "dP"  PiPs       ' 
A    —  +  ^        &    =0. 
~d*' P2pz     'dfi'pipz 
must  be  satisfied,  again  writing  _  =0. 
Thus  if  (/)  [and  i|r  are  two  functions  of  a  and  £,  we  may 
write 
~d&  ~d(f>  /kn 
a=?p*Ps*fip>      y=-pip-z^ai    •  ,  •    •    \P) 
whence  by  (;i), 
P1P2  ~~da'\  pi   3«/     3/3  \  pi  3£/  * 
Thus  with   (4),   writing    ™^2  =  —  &25  where    -r-  is 
ive-length,  V"  a*  * 
3  piP2(]$_(ih}h 3$\     J3  /i?2_£3 3£\  1   ,  PM  =n 
3/3  ^  ld«Vj>3  a«7  +  3£Vpi   30//        3/3    "  ' 
Similarly  from  (3)  and  (4) 
3_Pi£2  (  _d  //>i^3  B^\   (     3  (P2pz  3<M  1  +w 
3«  pB  \~da\  p2  3«/     3^\  pi  3£/ J        3« 
Thus 
P^=0. 
3  /^i/>3  3  jA      3  /j?2/?3  3ft\     gpg  ,  _0         {7) 
B*V>s  3J  +  3/3VPl  "bPJ^PipV*  '    K) 
By  the  symmetry  of  the  relations,  i|r  must  satisfy  the  same 
characteristic  equation. 
The  independence  of  the  functions  <£  and  i/r,  and  of  the 
corresponding  components  of  force  (x,  y,  c),  (a,  b,  z)  derived 
therefrom,  allows  of  the  extension  to  surfaces  of  revolution 
of  a  result  noticed  by  Macdonald  *  for  the  case  of  spheres. 
The  result  so  extended  may  be  stated  : — 
The  most  general  symmetrical  oscillation  possible  in  the 
space  between  two  surfaces  of  revolution  belonging  to  one 
of  three  orthogonal  systems  of  surfaces,  is  an  additive  com- 
bination of^two  types  :    (1)  a  type  in  which  the  components 
*  Electric  Waves,  Chap.  3. 
