704  Mr.  J.  W.  ^Nicholson  on  the  Symmetrical 
The  entire  investigation  is  confined  to  symmetrical  vibrations, 
which  are  the  most  interesting  from  a  physical  point  of 
view. 
The  electric  and  magnetic  vectors  are  first  expressed  in  a 
convenient  manner  in  terms  of  two  potential  functions 
satisfying  a  certain  differential  equation. 
Let  any  point  of  space  be  defined  by  (1)  cylindrical  co- 
ordinates (pooz),  where  z  is  the  distance  along  an  axis,  p  the 
perpendicular  upon  that  axis,  and  co  the  longitude  of  the 
meridian  of  the  point  :  and  (2)  coordinates  (afico),  where 
p+lz  =/(«+t£) (i) 
Then  the  surfaces  a  =  const.,  (3  =  const.,  &>  — const.,  form 
three  mutually  orthogonal  systems.  The  last  is  a  system  of 
planes,  and  the  others  are  surfaces  of  revolution  about  z. 
The  space  elements  normal  to  the  three  surfaces,  one  of 
each  system,  through  any  point,  are 
7  dot  d(S         7  dw  /OA 
dn1=  — ,     dn2=-J-,     dfio= — .    .      .      .      (2) 
Pi  P-2  Ps 
As  usual  (en/~),  (abc\  (uvw)  will  denote  components  of 
electric  force,  magnetic  induction,  and  current,  but  measured 
along  the  directions  whose  cosines  are 
(dnu  dn2,  d,n^)l(dn^  +  dn22  +  dn%2)*. 
The  magnetic  force  is  {a,  b,  <?)///,,  if  jx  be  the  permeability. 
For  all  investigations  here  contemplated,  fju=l. 
By  the  circuital  relation  expressing  Faraday's  law,  since 
-—  =  0  for  symmetrical  vibrations, 
—  a  _   ~d      £  b      _  ~d     _£ 
p?lh  ~  ~dfi  '  ps '  PiPs  ~  ~d*  '  ]h ' 
~'G  =  .B     v      B     x_  ^ 
P1P2     ~d"  ' l>.2      ~dfi  'pi 
where  dots  denote  differentiations  with  respect  to  t. 
The  appropriate  expression  of  Ampere's  law  leads  to 
—  z 
"P1P2 
c 
y             "d     c 
V2PiPs~     "d^ ' pz 
b        ~d      a 
>2          B/5  '  Pi  '    ' 
(4) 
writing  47rV2  (m,  v,  w)  =  (x}  y,  z),  where  V  is  the  velocity  of 
propagation  of  electromagnetic  disturbances  in  the  medium 
between  the  conductors. 
