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XXXVI.  On  Hyper  distributives.  By  Sir  James  Cockle,  F.R.S., 
Corresponding  Member  of  the  Literary  and  Philosophical  Society 
of  Manchester,  President  of  the  Queensland  Philosophical 
Society,  fyc* 
1.   T   ET  u  and  a  in 
-"  0(tO.+  0(fl)=0(t*+fl) (i) 
be  regarded  only  as  recipients  of  suffixes,  so  that  6{u)  is  a  func- 
tion, not  of  u,  but  of  independent  symbols  w0,  uv  w2,  . . . ,  and 
e(u)=e(u0,Ul,u2i..) (2) 
2.  Put 
^(W  +  «)=^(A0,A1,A2,..)       ...     (3) 
wherein 
A,  =  (u  +  a)r.     .     .     . (4) 
Then,  when  we  can  interpret  A,,  so  as  to  obtain  uniform  result?, 
the  function  6  may  be  called  a  hyperdistributive. 
3.  If 
Aw  =  wm  +  mvTO_1fl1+  ..  +am,        ...     (5) 
where  the  dexter  is  obtained  from  the  development  of  (u-\-a)m 
by  changing  exponents  into  suffixes,  uniform  results  may  be  ob- 
tained. 
4.  Let,  generally, 
0i(«)=«ii (6) 
02M=«2-< (?) 
B3(u)=:u3—3u1u^-\-2u^J (8) 
04(m)  =  h4— 4u,ti3— 3tt;  +  12t/«t£8-6ttJ;        .     .  (9) 
then  we  shall  have 
0I(«)  +  ^(fl)=^1(A), (10) 
*.(«)  + *.M=*.(A), (") 
03(u)+03(«)  =  03(A), (12) 
et(u)+e4(a)=e4{\), (i3) 
provided  that,  in  the  development  of  0(A),  we  replace  Ar  ly 
(u  +  a)r  and  then  change  exponents  into  suffixes. 
5.  These  results  are  all  comprised  in 
*..(«•) +  *.(«)  =  *.  (A) (14) 
"When  W2  =  2,  or  m  =  3,  the  results  of  the  last  article  may  be  ve- 
rified directly  without  any  great  labour.  When  m  =  4  the  cal- 
culation is  rather  longer;  but  much  of  the  work  is  already  dene 
*  Communicated  by  the  Rev.  Robert  Harley,  F.R.S. 
