172         Prof.  M.  B.  Pell  on  the  Constitution  of  Matter. 
whence  B=  — J0 ;  and  supposing  fi  =  l,  we  have 
cos(r-i)(9 
cos^0 
The  coefficient  of  «rw  in  the  resulting  equation  is 
/n        /i     i\/<      /•              2  sin  w0  sin  ^0 
(3  cos  fl-iy.-/.., ^T^' 
and  the  equation  is 
2 
2  sin  tz0  sin  i  0  _ 
J     a?  =0. 
cos  10 
In  a  similar  manner  we  obtain 
,  2sinrc0sini0  _ 
cos^0  l 
and  since  2  cos  0 .  #r=#r-i  +  a?r+u 
a?,.  =  Acos  (r0  +  B). 
The  first  equation  gives  B=  —  ^0,  and  therefore 
#V=Acos  (r—  J)0, 
a?!  =  A  cos  J  0, 
__  cos  (r— ^)0 
COs\Q  x 
If  we  put  ~  =y,  we  have 
sin  w0=  sin  0(2  cos  0— 2  cos  27)  (2  cos  0—2  cos  Ay)  . ,. 
(2  cos  0-2  cos  2(n-l)y)y 
2  sin  ?z0  sin  ^  0 
cosJ0 
=  4sin2i^(-4sin2^+4sin27)(-4sin2i^  +  4sm227)... 
And  since  2  cos  0=  ^(£j ' +  2}   -4  sin2  40  =  ^)*  the 
equation  for  determining  xx  becomes 
where  ^=2msin«yJ  5  being  any  number  from  1  to  7i— 1. 
doc 
If  -=7  =0  when  £=0,  for  all  values  of  r  we  have 
at 
x1=.a  +  alco^fiit+  . . .  +  as  cos  fjust  +  . .  .c/n^  cos  fin^i  tt 
