LAPLACE. 583 



Suppose then we have two unknown quantities, x and y ; we 

 find from (8) 



X = 





and the mean error for x will be . . 



The mean error to be apprehended for y may be deduced 

 from that for x by interchanging a^ with hi. 



If there are three unknown quantities we may deduce the 

 mean error from that which has just been given in the case of 

 two unknown quantities by the following rule : 



change ^a^ji into 'Za^^Ji ^o^'- > 



{tkco\Y 



cUi 



change l^h^^i into %KJi ^ 



change Xajjiji into Zapij\ — ^^ ' ^ ^T ' ""^ . 



To establish this rule we need only observe that if we have 

 three equations (8) we may begin the solution of them by ex- 

 pressing V from the last equation in terms of X and //., and sub- 

 stituting in the first and second. 



By a similar rule we can deduce the mean en'or in the case of 

 four unknown quantities from that in the case of three unknown 

 quantities : and so on. 



The rule is given by Laplace on his page 328, without any 

 demonstration. He assumes however the function of the facility 

 of error to be the same at every observation so that j\ is constant 

 for all values of i\ and he takes, as in Art. 1009, 



li! = 



2s ' 



1012. Laplace gives on his pages 329—332 an investigation 

 which approaches more nearly in generality to that which we 

 have sui3plied in Art. 1007 than those which we have hitherto 

 noticed in the fourth Chapter of the Theorie ...des Proh. ; see 

 Art. 917. Laplace takes the same function of the faciUty of error 



