^Klim  Mb  ancren  Mid  not  divUible  by  uiy  sqnate,  the  two  lum*

*-o  w

are  identical,  as  appears  from  a^  comparison  of  (2)  wUU  the  generalization  of  (A),  and  as
hu  been  already  observed  in  the  case  when  » is  a  prime  (art.  21).
1861.  z


33ff  RBPORT — 1861.
this  expression  the  sum  2(— 1)  '  (^)  is  zero,  because  the  terms  corre-
sponding to  m  and  2A+m  destroy  one  another;  so  that

-4AV=2.^<-^^VA2(-l)=^5)h(-S)-*|?]'

Distinguishing  the  two  cases  D=A,  and  D:=^A,  and  observing  that  the
imaginary  parts  vanish  identically,  as  they  ought  to  do,  because  V  is  real,  we
have,  finally,  if  D=A,

-4DV=2VDSC-I)'T^(5)logsin(^)


=2VD.g)logsin-;


and  if  D=- A,


2Va    \m/


(ii.)  The  series  2^— j-  can  also  be  summed  by  substituting  for  f — j  its

trigonometrical  value  deducible  from  the  formuIsB  of  Oauss.  We  will  take
as  an  example  the  case  in  which  Db  —  a^S,  mod  4.  Writing  n  for  m,  and
m  for  n,  in  the  equation

/(e^'0-2.^'n-l)'^(!)VA.
we  find,  observing  that  ^(l  +  A)  is  uneven,

the  summation  extending  to  every  value  of  m  prime  to  4A  and  leM  than  it
Substituting  this  expression  for  f  —  j  in  V,  we  have

SJVA    \m/    n      \  4A  /
Since  the  expression  which  we  have  substituted  for  f—j  is  zeroy  when  n  is
not  prime  to  4Ay  we  may  attribute  to  n,  in  the  series

either  all  uneven  values,  or  all  integral  values.    The  sum  of  the  series

sin  a:  .sin  Sar  ,  sin  5x_.
ON  THB  trasORT  OF  NUBCBEBS.  389

is»  by  a  known  theorem,  ^  or  — -^y  according  as  0<«<iri  or  ir<«<2r.
