LAPLACE. 567 



observation we shall denote it by f{z) ; and then dropping those 

 suffixes which are no longer necessary, we have 



h = [ V(^) ^^' ^' = f"*^!/ (^) ^-^^ 



J b ■ J b 



Such is the solution which we have borrowed from Poisson ; he 

 presents his investigation in slightly varying forms in the places 

 to which we have referred : we have not adopted any form ex- 

 clusively but have made a combination which should be most ser- 

 viceable for the object we have in view, namely, to indicate the 

 contents of Laplace's fourth Chapter. Our notation does not quite 

 agree with that which Poisson has employed in any of the forms of 

 his investigation ; we have, for example, found it expedient to 

 interchange Poisson's a and h. 



We may make two remarks before leaving Poisson's problem. 



I. We have supposed that the error at each observation lies 

 between the same limits, a and h ; but the investigation will apply 

 to the case in which the limits of error are different for different 

 observations. Suppose, for example, it is known at the first 

 observation that the error must lie between the limits a^ and h^, 

 which are within the limits a and h. Then f^ {z) will be a function 

 of z which must be taken to vanish for all values of z between h 

 and h^ and between a^ and a. 



Thus in fact it is only necessary to suppose that a and h are so 

 chosen, that no error at any observation can be algebraically greater 

 than a or less than h. 



II. Poisson shews how to proceed one step further in the ap- 

 proximation. We took y — Ix \ we have more closely y—lx — l^ , 

 where 



Hence, approximately, 



cos {y — ex) = cos {Ix — ex) + l^x^ sin {Ix — ex). 



