302  Sir  James  Cockle  on  Hyper  distributives, 
tial  coefficients  will,  after  all  reductions,  be  separated  from  a  and 
its  differential  coefficients.     Now,  when  in  the  development  of 
-i —  I  — r    there  occurs  an  expression    -  -= —  I,  then  there  occurs 
dxm  \u  dx)  ]  \u  dxnP 
in  the  dexter  of  (25)  the  corresponding  expression  ( ,  n    )  # 
But  if 
l=M-    •  •  (26) 
then,  by  Leibnitz's  theorem,  as  expressed  in  brackets, 
Tea    dxn  ua  {u}°{a}0' 
Make 
and  we  have 
{uy_       {«k 
{u}0-U»     {a}0 
(27) 
(28) 
ipFF  =  [a+ffl]" (29) 
10.  We  know  that  j-^{~  ~)  can  be  developed  in  terms  of  uu 
u2,  &c,  and  j-^  I  -  ~  J  in  terms  of  alf  #2,  &c.  The  article  pre- 
ceding shows  that  the  expression  for  ~j-^\- V-M  can  be  ob- 
tained from  either  development  by  writing  therein  [w  +  a]rin 
place  of  ur  or  ar  respectively.  This  is  the  same  thing  as  repla- 
cing ur  or  ar  by  A,.. 
11.  Let,  then, 
dxm 
and  0m(u)  will  be  a  hyperdistributive,  and  may,  I  think,  be  called 
the  primary  hyperdistributive  of  the  mth  order.  The  symbols  ul} 
Vq,  . .  wmare  independent;  for  u  is  an  arbitrary  function  of  x.  As 
observed  by  the  great  De  Morgan  (Camb.  Trans,  vol.  ix.  part  2), 
undefined  dependence  is  independence. 
12.  We  may  denote  the  nth.  power  of  |w}rby  {w}r(n),  where 
r  is  the  algorithmic  index,  and  n  or  (n)  an  ordinary  exponent. 
This  notation,  however,  will  not  be  much  needed  in  what  follows; 
for  we  can  replace  algorithmic  indices  by  suffixes. 
