358  Mr.  J.  Morrow  on  the  Lateral  Vibration 
In  the  second  approximation, 
-^=4-5324^; 
y\  pi- 
and  in  the  third  approximation, 
-y  =  '69|^ri(-304931  P— 874742  l*x  +  -957615  ZV 
-•542880  «V  +  -192,901  ZV-  '044091  /^  +  '006889  <*6 
- -000656  j-+ -000033  ^\ 
.Vi       ,  nn.EA2  a     a-     2-170  AU 
—  —  =4wl0b — pr,     and     IS  =  — -■ -. 
V\  Pl~  2tt    L  ' 
Kirchhoff  *  has  obtained  the  solution  for  the  frequency  in 
this  case,  and  his  result  may  be  written 
_  2-179  ATJ 
~    2tt       I   ' 
In  the  Euler-Bernoulli  theory  of  beams  it  is  assumed  that 
the  greatest  diameter  is  small  compared  with  the  total  effective 
length.  When  the  diameter  varies  as  the  distance  from  one 
end,  therefore,  it  is  necessary  that  the  variation  should  be 
small.  In  other  cases,  so  far  as  the  mathematical  theory  is 
concerned,  the  solution  can  only  be  looked  upon  as  a  probable 
approximation. 
Section  III.  Loaded  Bars  of  Negligible  Mass. 
§  6.  When  the  bar  carries  a  load  at  some  point  in  its  length, 
and  the  mass  of  the  bar  itself  is  negligible,  it  is  not  neces- 
sary to  assume  a  type  of  displacement.  The  method  then 
becomes  an  exact  one,  and  gives  at  once  the  true  type  and 
frequency  of  the  vibration. 
In  the  following  paragraphs,  m  is  the  mass  with  which  the 
bar  is  loaded,  and  a  is  the  distance  of  its  centre  of  gravity 
from  the  point  chosen  as  origin. 
§  7.  Clamped-Free  Bar. — If  the  origin  be  at  the  load  and 
y1  be  the  displacement  there, 
™T  d2y 
and  the  vibration-curve  is 
*  See  Berliner  Monatsbenchte,  1879,  p.  815. 
