174         Prof.  M.  B.  Pell  on  the  Constitution  of  Matter. 
equation  (4), 
jl  i  \      l  ,  yet,       cos(2r  —  l)sy 
$i  (r)  =  b  +  2,bsfjbs s £-£- 
^iV  '  5™        cossy 
These  equations  give,  as  before, 
*£?*«  cos  (2r-l)Sy=5^^, 
Mnw        v  '  '      2cossy 
+  3 S::r,2^,WcoS(2,-l)S7)  "^dnM     (7) 
n    s~l         v  '  /^ 
which  completes  the  solution  to  a  first  approximation. 
It  may  be  observed  that  the  formulae  given  by  Poisson  for  the 
longitudinal  vibrations  of  an  elastic  rod  may  be  easily  deduced 
from  the  above  results.     It  is  remarkable  also  that,  by  putting 
£  =  0  and  ,vr  =  (j>(r)  in  equation  (6),  a  general  analytical  theorem 
may  be   deduced,    of  which  Lagrange's   theorem,    that  when 
<£(0)=0  and  <£(«)  =  (), 
, ,  N       2  ~oo  /  Ca ,  r  n    •    nirx\  .    nirx 
*(*)  =  SS.  (J^(*)«n— )«*  — 
is  a  particular  case. 
In  order  to  determine  how  the  system  would  vibrate  if  dis- 
turbed and  then  left  to  itself,  suppose  the  first  atom  to  receive 
a  blow  impressing  upon  it  a  velocity  ma,  which,  if  the  next  atom 
were  fixed,  would  cause  it  to  vibrate  through  a  space  a  nearly, 
a  being  small  compared  with  h.  We  have  then  <b(r)  =0, 
^)j(r)  =ma  when  r=l,  and  zero  for  all  other  values. 
2^>j(r)  cos  (2r— l)sy— ma  sin  sy, 
%<l>i{r)  =ma, 
mat      a  ~s=n-i  .    /n       1N       ■        . 
xr  — h  -  2    ,     sin  (2r—  l)sy  sin  fiat. 
n        n    s=1 
The  nature  of  the  vibrations  will  be  the  same,  and  the  term 
— ,  denoting  the  general  motion  of  translation  of  the  system, 
n 
be  avoided,  by  supposing  the  initial  conditions  to  be 
<£(r)=  ^sin(2r-l)5y,     ^W=0; 
then 
a>  =  ~^s~_n~1sin(2r— l)sycos/j,st. 
