34 
ME.  A.  CATLET  ON  THE  THEOET  OE  MATEICES. 
48.  The  convertible  matrices  may  be  given  in  the  first  instance  in  the  ordinar}"  form, 
or  we  may  take  these  matrices  to  be 
( a,  ^ V ),  ( y ) &c. 
c,  d 
c',  d' 
c",  d"  ! 
where  of  course  d — a : b : c=d''<—a' : V ; d=dj'  — a" : b" : c"=&c.  Here  writing  h—d—a, 
and  consequently  11=  — {d — 4Jc,  and  assuming  also  and  cot  and 
V R 
in  like  manner  for  the  accented  letters,  the  several  matrices  are  respectively 
(iv/E.  q)  (iv/E,  s'),  (is/E,  /),  &c., 
and  the  compound  matrix  is 
- S+S'+S"+  ■ ■)  • 
49.  When  the  several  matrices  are  each  of  them  equal  to 
« 
( «,  5 ), 
\ c,  d\ 
we  have  of  course  g'=g''=5'"..,  E=Il'=E"..,  and  we  find 
\ c,  d \ \ ^ / 
or  substituting  for  the  right-hand  side,  the  matrix  represented  by  this  notation,  and 
putting  for  greater  simplicity 
we  find 
S^'(iyS)-=(iyR)L,  orL=S^»(iyE)-, 
( «,  5 )"=(iL(\/Ecotw5'-— (c?— a)),  U 
) 
c,  d 
Lc 
,-|L(\/E  cot  nq-\-{d— a)] 
where  it  will  be  remembered  that 
d-\-  d 
— and  cotq= 
the  last  of  which  equations  may  be  replaced  by 
cos^-fy-lsm^=^^y=^- 
The  formula  in  fact  extends  to  negative  or  fractional  values  of  the  index  and  when  n 
is  a fraction,  we  must,  as  usual,  in  order  to  exhibit  the  formula  in  its  proper  generality, 
write  §'-|-2niT  instead  of  q.  In  the  particular  case  n=-\,  it  would  be  easy  to  show  the 
identity  of  the  value  of  the  square  root  of  the  matrix  with  that  before  obtained  by  a 
different  process. 
