Notices  respecting  New  Books,  305 
will  remain  unaltered  when,  having  replaced  ar  by  [#  +  w]r,  we 
substitute  for  ur  its  value  as  given  by  (44).  If  we  assume  u  to 
be  a  rational  and  entire  function  of  x,  its  degree  should  not  be 
less  than  m,  or  our  form  will  be  needlessly  restricted.  We 
thus  obtain  the  entire  criticoidal  functions.  The  symbols  a  and 
A  are  not  in  general  subject  to  any  relation  corresponding  to 
(44).  I  say  entire  critical  and  criticoidal  functions,  because 
there  are  other  such  functions,  as  will  be  seen  on  reference  to 
my  paper  "On  Fractional  Criticoids"  in  the  last  May  (1871) 
Number  of  this  Journal*. 
21.  All  functions  of  critical  or  criticoidal  functions  are,  re- 
spectively, critical  or  criticoidal.  And  critical  functions  and 
criticoids  are  thus  seen  to  be  not  merely  connected  by  the  ana- 
logies of  algebra  and  the  calculus,  but  to  have,  in  hyperdistri- 
butives,  a  common  algorithmic  origin. 
"  Oakwal "  near  Brisbane,  Queensland, 
Australia,  January  23,  1872. 
XXXVII.  Notices  respecting  New  Books, 
Monthly  Notices  of  the  Royal  Astronomical  Society.  The  President's 
Address  on  the  Presentation  of  the  Gold  Medal,  Feb.  1872. 
Observations  of  Comets  from  B.C.  611  to  a.d.  1640.  Extracted  from 
the  Chinese  Annals  by  Johx  Williams,  F.S.A.     London,  1871. 
AMIDST  the  eager  pursuit  of  information  capable  of  throwing 
additional  light  on  the  physics  of  the  sun,  it  is  refreshing  to 
find  the  Royal  Astronomical  Society  directing  its  renewed  attention 
*  Phil.  Mag.  S.  4.  vol.  xli.  No.  2/4,  pp.  360-368.  Some  of  the  fore- 
going formulae  involve  questions  of  partition.  The  following  notation  of 
partitions  seems  to  be  convenient.     Put 
and  let  ii(n)  denote  the  number  of  different  partitions  of  n,  including  n 
itself.  Also  let  irm{n)  denote  the  number  of  different  partitions  of  n  into 
m  parts,  no  zero  part  occurring.  Let  (f>(n)  denote  the  number  of  different 
partitions  of  n  in  which  no  unit  part  occurs.  Also  let  <f>m{n)  denote  the 
number  of  different  partitions  of  n  into  m  parts,  no  unit  or  zero  part  oc- 
curring.    Then 
n{n)  =  S„7rr(n) ;     #(n)  =  Sn$r(w) ; 
<^m(n  +  m)  =  7rm(n)  =  Sm7rr(»— m); 
n (2n)  =  1+  S*7r(r)  +  S„_  l7r r  (2») ; 
ir(2n  +  l)  =  l  +  S»{7r(r)  +  7rr(2»+l)}; 
and  consequently 
7rm(2m)  =  7r(w). 
Thus 
7r(4)  =  l  +  7r(2)  +  7r(l)+7r1(4)  =  5 
exemplifies  the  notation. 
Phil,  Mag.  S.  4.  Vol.  43.  No.  236.  April  1872.  X 
