for the Theory of Errors. 487 



minations by the method of double weighing, we must 

 compound this curve with another equal and similar curve. 



Fiff. 11. 



9. Most advantageous combination. — Suppose we have a 

 number (n) of independent measurements (a 1? a 2 , . . . a n ) of 

 the same quantity. Let the chance of an error between x x 

 and x 1 + dx; l in a x be 



1 ^:. _fcW «fai, (3) 



with similar notation for the remaining measures — an assump- 

 tion which is perfectly general. By adopting X as the 

 true value we assume the errors in the respective measures 

 to be Oh-X), («,-X), . ... K-X). 



If we know the limits of possible error for one of the 

 measures, a r , then a r — X must be made to fall within these 

 limits. Subject to such restrictions we must make the 

 product 



1 _<p( a X) 1 -<?5 2 (a 2 -X) 1 -<p ft (a -X) 



e ~ e ... z= . e 



C lv /7r C 2V /7T C n V ' 



a maximum ; that is we must make 

 $i( a i — X) + </> 2 (a 2 — X)+ . .. -\-<f>n(a n ~ X) a minimum . (4) 



If Laplace's law of error applies to each of the independent 

 measures, then 



Jk4» ^\ ("I ~ X ) 2 jl („ YA- (^~^) 2 



9i( a i~" A ) — 7^2 — ? 9n(a»— A J— — pi — . 



and (4) reduces to the method of least squares. We can also 

 see from (4) how the most advantageous combination of 

 measures depends on the laws of error in the separate obser- 

 vations ; and that Laplace's law of error alone leads to the 

 method of least squares. 



As another particular case, let each of the quantities 

 </> 1 (^ 1 ), $2(^2) • • • <t>n(%n) be constant between certain limits, 

 and possible only between those limits ; this will in general 

 give a finite possible range to the value of X, all values 

 within this range being equally probable. 



