194         Mr.  J.  W.  L.  Glaisher  on  Certain  portions  of 
Angle 
j  of  plates. 
o 
P 
*+p' 
1st        nir»«p  tri  flip  limh     - 
35 
•158 
("At  about  10'  distance  from  the  limb 
45 
•275 
40 
•212 
2nd.-] 
45 
•275 
l.                       a                         a 
45 
•275 
Mean     . 
•239 
Arconum,  January  27,  1872. 
XXII.  Remarks  on  certain  portions  of  Laplace's  Proof  of  the 
Method  of  Least  Squares.  By  J.  W.  L.  Glatsher,  B.A., 
F.R.A.S.,  Fellow  of  Trinity  College,  Cambridge  *. 
A  CONSIDERABLE  portion  of  the  fourth  chapter  of  La- 
place's Theorie  des  Probabilites  is  devoted  to  the  investi- 
gation of  the  law  of  facility  of  error  of  the  mean  &c.  of  a  great 
number  of  observations,  all  the  errors  of  which  are  subject  to 
the  same  law  of  facility  <£(#) .  Laplace,  as  is  well  known,  obtains 
his  result  by  the  consideration  of  the  coefficient  of  e1™1  in  the 
expansion  of 
{,g)e--+,(»^)e---..+,g)...+^g)e-}: 
A  great  simplification  of  this  part  of  the  analysis,  with  increase 
of  generality,  was  effected  by  Leslie  Ellis,  who  stated  the  pro- 
blem in  the  following  manner  f  :  to  find 
subject  to  the  condition 
/i1e1+^2€2...  +  ^^=w. 
This  multiple  integral  Ellis  evaluated  approximately  by  writing 
for  6n{en)  its  equivalent  $n( ^  1  '  " —         n~),   replacing 
this  last  function  by  the  double  integral  of  Fourier's  theorem 
and  taking  all  the  integrals  between  the  limits  +  go  .  Integra- 
ting the  result,  with  regard  to  u,  between  — /  and  /,  the  proba- 
bility of  2)fte  being  intermediate  in  magnitude  to  these  two 
quantities  is  found.  The  object  of  the  present  communication 
is  to  obtain  the  usual  result  by  a  method  which,  though  bear- 
ing a  strong  resemblance  to  Ellis's  investigation,  nevertheless 
seems  to  reader  the  analysis  more  elegant  and  symmetrical ;  the 
*  Communicated  by  the  Author. 
t  Cambridge  Philosophical  Transactions,  vol.  viii. 
