304  Sir  James  Cockle  onHyperdistributives. 
butive.  Moreover  if  fm{u)  be  any  distributive  function  of  any 
quantity  u,  and 
/-(«)+/.W-»*M.  .._..-  (38) 
then  §m(u),  or  any  linear  function  of  such  functions,  is  hyper- 
distributive.  The  suffix-recipient  u  in  0(u),  and  the  explicit 
quantity  or  variable  u  in  f(u),  are  independent,  and  have  no 
connexion.  But  we  may  if  vye  please  establish  a  counexion  be- 
tween uv  u2, . .  &c,  and  between  them  and  the  u  in/(w). 
19.  Thus,  lety(w)  vanish,  and  suppose  that,  for  all  values  of  r, 
[My=(M)'ww; (39) 
Then 
U,.=  [w-Ml)r=K-Wi)r=0.  .  .  (40) 
Hence  (14)  will  take  the  form 
*.(«)  =  *«  (A) (41) 
the  meaning  of  which  result  is,  that  0m(a)  will  remain  unaltered 
when,  in  it,  we  replace  ar  by  [a  +  u)rf  or  its  equivalent  (u  +  «]'• 
When  (39)  is  satisfied,  (14),  becoming  (41),  gives  us  the  entire 
critical  functions.  It  must  be  borne  in  mind  that  (39)  does  not 
imply  any  such  relations  as 
[a]r=  (fl)r  =  ar=fll .     (42) 
20.  Again,  let 
/»M=/-M  =  -^pk (43) 
and,  for  all  values  of  r,  let 
Ur~udx-' 
(44) 
Then 
8n(uvuq,...u^-€iI-^L=0,       .     .     .     (45) 
whatever  u  may  be,  and  the  hyperdistributive  relation 
w, 
0m{uu  Kg,  .  0-     ^m-l     +0™(al>  **•*)- 
=MA,,A^.  •)-£#■     ....•••     (46) 
becomes 
«.(«.,  ^  ..«-)■*  ^  ='-(*„  A» . .  A.)  -  =p£l  •    (47) 
jm-l 
and  the  meaning  of  this  result  is,  that  0w(rti>  #2, . .  am)  —  t-j^J 
