718  Mr.  J.  W.  Nicholson  on  the  Symmetrical 
The  high  periods  of  a  single  spheroid  are  given  by 
(1)        C?^X    p(P>codb««-n.n  +  l)Me=0,     .     (57) 
d    r  \/smha 
W        Ja  1  (Fc2cosh2*-n.n+T)i 
X  ^  |  I  *(AV  cosh2  cc-n.n  +  1) Ua X  =  0,      .      (58) 
where  the  upper  or  lower  functions  are  taken  according  as  n 
is  odd  or  even.  These  give  the  high  vibrations  of  the  singly 
infinite  system  defined  by  n. 
The  investigation  given  in  a  previous  paper  *  for  the  elliptic 
cylinders  becomes,  if  stated  in  full  detail,  exactly  similar  to 
the  present.  The  ft  factor  is  restricted  in  a  similar  way,  but 
its  adjustment  for  uniformity  on  the  axis  is  incorrectly  given. 
The  true  value,  in  the  notation  there  employed,  oscillates 
between  /r~ i  cosh  k  \/b2-\-  fj,  and  /jl~ i  sinh  h  V^  +  A6  accord- 
ing to  the  value  of  n.  The  equation  there  numbered  (51) 
ought  to  be  written 
ffi}(v^-;)}-o,  . 
TT 
according   to  the  value   of  n.      The  —  is   determined  most 
°  1 
readily  from  the  corresponding  Bessel  function. 
Series  of  Periods  corresponding  to  High  Values  of  n. 
The  solution  may  be  expressed  approximately  in  the  general 
case  when  n  is  large,  even  though  kc  may  not  be  so  large 
that  the  value  of  X  is  (kc)2.  Writing  X  =  n  .  n  + 1  +  X',  or, 
when  n  is  large,  X  =  ;r  +  X/,  the  asymptotic  expansion  of  the 
equations  for  <£  when  n  is  large,  found  just  as  in  the  case 
when  kc  is  large,  lead  to 
Acos(G>J+61)cos(o)2  +  e2)cosArV«      r^r t—=  rcLKS 
$=- ^y v/smhasm/3     .     (bO) 
where 
«,'=— l=(€2eosh2«-X,-«2)i.     .     .     .      (61) 
o),'=^2=(^  +  X'-e2cos2/3)i      .     .     .     (62) 
da. 
*  Zoc.  cit. 
