LAPLACE. 571 



where Z = S I (j>ii^)fi (^) ^^' 



'a 



b 



2 



and 2«=' = 2 j ^ U (2)| f, iz) rf.t! - 2 1 j^, [z)f, {z) rf.j . 



The summations extend for all values of i from 1 to s, both 

 inclusive. 



It is not necessary that </)i {£) should be restricted to denote 

 the same function of z for all the values of i\ Poisson however 

 finds it sufiicient for his purpose to allow this restriction. 



Suppose now, for example, that cf)i{s) = z" for all the values 



of ^; and let the function of the facility of error be the same 



at every observation. Then, taking 6 = 0, as in the preceding 



Article, 



ra 

 1= s j z^f {z) dzj 



'a { ra 



2/c' = 5J z'f{z)dz-sU zy{z)dz)-. 



Take c = l; then the probability that the sum of the squares 

 of the errors will lie between l — rj and I + 7] is 





"~ dv. 



This will be found to agree with Laplace's page 312. 



1007. Laplace proceeds in his pages 313 — 321 to demonstrate 

 the advantage of the method of least squares in the simplest case, 

 that is when one unknown element is to be determined from 

 observations; see Art. 921. This leads him to make an investi- 

 gation similar to that which we have given in Art. 1002 from 

 Poisson : Laplace however assumes that the function of the facihty 

 of error is the same at every observation, and that positive and 

 negative errors are equally likely, and thus his investigation is 

 less general than Poisson's. 



Laplace and Poisson agree closely in their application of the 

 investigation to the method of least squares : we will follow the 

 latter. 



