Sir  James  Cockle  oa  Hy <per distributives,  303 
13.  We  know  that 
■~=um+i-ulum (31) 
And  hence  we  can  deduce 
i-[w-W])m==[w--Wl)-+i--m[w-w1)2[w-Wi)wl"1.    ■   '(32) 
Let  us  put 
[u-iH)m=Vm (33) 
and  (32)  becomes 
^  =  U„i+1-mU2Um_1 (34) 
14.  Recurring  to  (30),  we  see  that 
and  the  dexter  of  this  result,  regarded  in  the  light  of  (34),  sug- 
gests that  0m  (u)  may  be  expressed  in  terms  of  Ug,  U3, . .  XJm . 
The  quantity  Uj  vanishes  identically. 
15.  Up  to  the  eighth  order  inclusive,  I  find  the  hyperdistri- 
butives  as  follows  : — 
02(W)=U2,     08(10  =  17*     04(W)=U4-3U*, 
eb{u)  =u5-iou2u3,   06(tt)=u6-i5Uau4-iou;+3ou!!. 
6>7(w)=U7-21U2U5-35U3U4  +  210UlU3, 
6Q(u)  =  U8-28U2U6-  56U3U5-35U*  +  70U2(6U2U4 
+  8U1-9UJ)- 
16.  If  we  denote  by  Vm  the  terms,  or  the  aggregate  of  the 
terms,  in  6m{u)  which  involve  triple,  or  higher,  partitions  of 
m,  all  the  foregoing  results  will  be  comprised  in  the  formula 
0»=2Um-i[U  +  Ur+P„  ....     (36) 
17.  By  way  of  example, 
U4=[w—w1)4=v4— 4w1w34-6^w2— 41/Jwj  +  mJ, 
3Vl  =  3[u-uiyw  =  3(uz-2ulul  +  uy  =  3(uz--ii'l)*. 
Hence,  developing  the  last  exprestion  and  subtracting, 
U4-3Ll  =  w4-4w,v3-3w2+12w'z/2-6i4,       .     .     (37) 
which  agrees  with  (9). 
18.  All  linear  functions  of  hyperdistributives  are  hyptidistri- 
