Sir  James  Cockle  on  Hyperdistributives.  301 
in  my  paper  "  On  Criticoids ,J  in  the  Number  of  this  Journal  for 
March  1870.     If  we  put 
-jr{u)  =  1/4—4^1/3  +  3*4       ....     (15) 
then 
+(A)  =  +  (a)  +  \20,(a)e2(u)+  +  (u),       (10) 
whence,  subtracting  six  times  the  square  of  (11), 
+  (.\)-6{83(A)\*  =  +(a)-6{0,(a)l*  ++(;u)-6{e2(u)\*,   (17) 
which  reduces  to  (13). 
6.  Marking  transitions,  analogous  to  that  from  exponents  to 
suffixes,  by  special  brackets  j.J-,   [.],    -j.],  &c,  I  reserve  the 
parenthesis  (.)  for  ordinary  involution.  The  common  expansion 
of  {u  +  a)m  may  be  written  in  either  of  the  forms 
{u  +  a)m={u)m  +  m(u)m-\ay  +  /.+{a)m,      .     .     (18) 
(u  +  a)m=(a)°(u)m  +  m(a)\u)m-l+  . .  +  («)»°.     (19) 
7.  Following  up  the  latter  form,  I  shall  put 
\u  +  a']m=[a]o{u}m  +  m[ay{u}m-1+  kc,   .     .     (20) 
the  convention  being  that  the  symbol  contiguous  to  a  bracket 
in  the  undeveloped  form  is  affected  by  that  bracket  in  the  deve- 
lopment. There  are,  of  course,  corresponding  developments  of 
[u  +  a\m,  \u  +  a\m,  [w  +  «]m,  and  of  {u  +  a)m,  (u  +  a]m,  &c. 
The  symbols  \u\r  are  arbitrary,  or  may  have  one  meaning  and 
be  connected  by  one  law,  while  [a]r,  and  indeed  \a\r  are  also 
arbitrary,  and  may  have  the  same,  or  another  meaning,  and  be 
connected  by  the  same  or  another  law. 
8.  Let  it  be  a  property  of  the  square  bracket  [.]  that 
[«]""=«», (21) 
and  that 
[w]o  =  Wo=,l, (22) 
in  the  same  way  that  for  the  parenthesis  (y)0  =  y0=l.     Then 
{u+ay*={u}m+mal{u}«-1+..+am{u}oj    .     (?8) 
and  (4)  becomes 
Ar=[u  +  aY (24 
9.  We  can  now  obtain  general  forms  of  hyperdistributives. 
An  inspection  of  the  identity 
dxm\u  dxr  dxm\a  dx)     dxm\ua    dx)       '     >J' 
shows  that,  in  the  development  of  the  dexter,  u  and  its  differen- 
