Prof.  M.  B.  Pell  on  the  Constitution  of  Matter.         173 
where  a,  ax, .  • .  as  are  arbitrary  constants.     It  may  be  easily 
shown  that  a  is  the  same  for  all  values  of  r,  since 
_  cos  (r  — £)0 
Xr~       cos  4,0       °°li 
and  that,  when  operating  upon  cos  fj,st,  0  =  2sy  ; 
,  ««=»-!     cos  (2;-  —  l)sy  /ON 
#,.=0  +  2,.    ffs —cos  fist.      .     :     (3) 
5-1  cossy  r  v  ' 
If  #,.=0  when  ^=0,  for  all  values  of  r  we  find  in  the  same  way 
..  ,  «*=«-a.  cos  (2r  — 1)57   .  ... 
s-l  COS  57  s  W 
r  being  the  number  of  the  atorn,  s  the  number  of  the  term   in 
a\,  and  b,  bv  . . .  arbitrary  constants. 
Suppose  the  initial  conditions  to  be 
where  <£  is  of  any  given  form.  For  the  determination  of  the 
arbitrary  constants,  we  have  n  equations  of  the  form 
...              vs=n-l      cos  (2r— 1)57  ,  . 
6(r)=a  +  Z_.     as > - — .      ...      (5) 
T\  i  s-\         s  COS  S7  V    ' 
It  may  be  shown  that,  if  p  and  q  be  any  two  integers, 
^r=i  cos  (2^  — l)^y  cos  (2r— l)gy  =  0, 
71 
except  when  p  =  q,  when  the  snm  is  -.     If,  then,  the  equations 
of  the  form  (5)  be  multiplied  respectively  by  cos  57,  cos  357, . . . 
cos  (2r  — 1)57.  . .  and  added  together,  all  the  terms  on  the  right- 
hand  side  disappear  except  those  involving  aSJ  and  we  have 
^Jltir)  cos  (2r-l)sy= j  n> 
cossy 
and  adding  together  equations  (5)  as  they  stand, 
%<l>{r)=na, 
.*.  a:  =  -2<£(r) 
n     T    ' 
+  -  ^I""1  S^(^(r)c08(2r-l)«y)  cos  (2r- 1>7  cos  pjt.     (6) 
dx 
If  the  initial  conditions  be   #r=0,  -y -=<f>[{r),  then,  from 
at 
