288        Lord  Rayleigh  on  the  Production,  of  Vibrations 
If  the  force  operates  only  between  t  =  0  and  t  =  t  and 
we  require  the  value  of  cf>  at  a  time  t  greater  than  t,  that  is 
is  after  the  operation  of  the  force  has  ceased,  we  may  write 
0  =  -  P  sin  n  [t-t')  ®t=t>  dt'.    .     .     .      (16) 
11 J  o 
If  r  be  infinitely  small,  the  force  reduces  to  an  impulse, 
and  we  get 
(j>  —  n~l  sin  nt . §  <f>  dt  ;       .     .      .      .      (17) 
but  it  is  the  other  extreme  which  concerns  us  at  present. 
In  many  cases,  especially  when  <£  =  0  at  the  limits,  we 
may  advantageously  integrate  (16)  by  parts.     Thus 
$  —  n~2  4>/l=r  cos  n  (t  —  r)  -  n~2  <J>*=0  cos  nt 
-i  {Tcosn{t-t')~dt'.    .     .     (18) 
n-   )  0  dt 
Again 
1       P"  ,  M^ 
-  -2       cosm    *-f )-J-7t 
n2J0  dt 
u,l   <m^smn(t-T) 
iv'      dt  y        ; 
1  d<Pf=0 
n*     dt     ^nt-lsfannit-t^dt',        (ID) 
and  so  on  if  required.  In  this  way  we  obtain  a  series  pro- 
ceeding by  descending  powers  of  n,  and  thus  presumably 
advantageous  when  n  is  great. 
As  an  example,  let  <£  =  t,  so  that  d2  <£  /  dt2  —  0.  The 
force  rises  from  zero  at  t  =  0  to  a  greatest  value  at  t  =  t 
and  then  suddenly  drops  to  zero.     From  (18),  (19)  we  find 
cj)  =  n~2  r  cos  n  (t  —  r)  4-  n~3  sin  ?i  [t— t)  —  n-3  sin  rc£.      (20) 
Again,  take  the  parabolic  law 
3>  =  /t  -  t2, (21) 
so  that  <£  =  0  at  both  limits, 
d®fdt  =  T—2t,  d<P/dt2  =  -2. 
From  (18),  (19) 
</>  =  —  rn~3  sin  n  (t  —  t)  — t  n~3  sin  nt 
+  2f/-4cos  //.  (t  —  t)  —  2n-4  cos  nt 
=  —  2n~3  t  cos  |wt.  sin  ?*  (£  — j>  t) 
+  -l^-4sin^?iT.  sinw  (£  —  |t).       .     (22) 
