Dielectric  Strength  of  Air.  253 
1/y,  we  find  that 
,=^r1+i  +  !2+9+n  +  33 
J      *    \      y    y2    f     y     f 
M  +  1W  +  ?M+-      1 (18) 
f      y       y  > 
It  will  be  seen  that  the  coefficients  o£  1/y  are  rapidly  getting 
larger,  but  it  has  to  be  remembered  that  /  must  equal  unity 
when  y— 2  is  zero.  We  therefore  alter  the  above  formula  so 
as  to  make  f—1  when  y  is  2,  and  yet  make  the  expanded 
form  of  the'  altered  formula  agree  with  (18)  as  far  as  the 
coefficient  of  1/y8.  By  this  means  we  secure  that  the  formula 
(19)  gives  the  correct  value  of/  when  y  is  2,  and  again  when 
we  can  neglect  the  ninth  term  in  the  series  formula  (15). 
Expanding  #/(;/— 2)  in  powers  of  1/y  as  far  as  the  term  con- 
taining the  eighth  power,  and  substituting  in  (18)  we  get 
'— T"l     y    y-2    y3    y4     f    y1     y*f 
approximately,  or 
/=to_1)+!+fc2{JF  +  l  +  I-?i-i,}..(ls> 
Substituting  2  +  x/a  for  y,  we  get 
/_  if   I       \\  1         •        x/a  •   _     x\a 
"*"  2{x/a  +  2f       (x/a  +  2)7       {x/a  +  2)8'     *     *  ^> 
The  values  of /are  easily  computed  by  this  formula.  For 
values  of  x/a  less  than  0*1  or  greater  than  0*7  the  error  is 
less  than  1  in  1000,  whilst  for  values  of  x\a  between  0'1  and 
0*7  the  error  is  never  as  great  as  2  in  1000.  For  practical 
purposes  therefore  the  formula  (20)  gives  the  values  of /with 
sufficient  accuracy.  We  could  have  made  it  more  accurate 
by  taking  more  terms  into  account  in  the  expansion  (18), 
but  we  have  not  done  so,  as  we  have  found  by  actual  com- 
putation that  the  Kirchhoti-Schuster  formula 
,_,      1  v  .    1    x2         73      x* 
'"        3'a  "t"45'a2      53760V       *     *     '  ["> 
sums  the  series  with  a  most  gratifying  accuracy  until  xja  gets 
greater  than  0*7.  Jt  therefore  completely  covers  the  part  of 
the  scale  of  our  formula  which  is  slightly  inaccurate.  The 
formulo3  (20)  and  (21)  therefore  give  the  complete  practical 
