Prof.  M.  B.  Pell  on  the  Constitution  of  Matter.         181 
when  n  is  large.     The  maximum  temperature  is 
4cos2(2?i  +  l)<f* 
There  is  thus  a  concentration  of  heat  at  the  joints ;  so  that  at 
those  points  a  greater  softening  takes  place  than  would  occur  if 
the  heat  were  uniformly  distributed.  This  effect  is  increased  as 
the  temperature  increases,  on  account  of  the  increasing  propor- 
tion of  the  heat  employed  in  softening  the  joints,  which  becomes 
latent.  The  temperature,  therefore,  at  which  the  melting  occurs 
is  much  lower  than  what,  if  uniformly  distributed,  would  dissever 
the  atoms. 
The  effect  of  radiant  heat  upon  a  system  of  n  atoms,  such  as 
that  under  consideration,  may  perhaps  be  represented  by  sup- 
posing the  additional  atom  at  the  beginning  of  the  series  to  be 
of  feeble  power,  proportional  to  \m~,  when  X  is  very  small.  Let 
x0  be  the  displacement  of  this  atom,  and 
#0  =  fi5  cos  fit ; 
the  equation  of  motion  for  the  first  of  the  n  atoms  is 
d2x  ' 
—  m2  (#2 — xx )  +  Xmq  (#o  ~"  x\ )  > 
dt* 
or 
1    d2x 
n7'aW"+  ^  +X^'1  =X<2?o  +  *2 ; 
neglecting  X  in  comparison  with  1,  and  putting 
we  have 
(q  — 1)^  =  ^2  +  X#0> 
qXc£=-X\  -f- x3) 
(q  —  l)xn  =  xn.l. 
These  equations  give 
2smn0sm±0         ,         N  ,  cos(?z  —  r  +  $)d 
r7T-^~  xn  =  \x0  =  Aa  cos  at  xr= r-^ x 
cos^0  n         °  cos\6 
\a  cos  (n — j'  +  ^)6  cos  at  _ 
2  s'mnd  s'm^O 
and  the  relative  displacement  is 
\a  sin  (n—r)6  cos  at 
(8) 
?r  +  l  —  X, —  — 
iin  nd 
