Results of Experiments. 423 



When, as occasionally happens, x l7 x 2 , x z are in arithmetical 

 progression with a common difference h, 



_ A% 



'^i- oi 8 ' 



Owing to experimental error the differences found are never 

 exactly equal, and Mr. J. Hopkinson (Messenger of Mathe- 

 matics, 1872, ii. p. 65) proposes to add together the various 

 values obtained and take the mean as the most probable value. 

 Though the method has no theoretical basis to rest upon, it is 

 comparatively easy and should give the same results in all 

 hands, while any method of plotting and curve drawing intro- 

 duces a further series of observations liable to personal error. 

 There are three great advantages : — 



(a) The sum of the differences of the observed and calcu- 

 lated values = 0. 



(b) If the values in the last column of differences regularly 

 increase or decrease, another term must be added to the 

 equation. 



(c) If the differences are very irregular, there is a want 

 of accuracy in the observations, or the theoretical equation is 

 carried further than the experiments warrant. 



If N be the number of observational equations each 

 differencing removes one, so that going as far as S 3 u there 

 are really only N — 3 = rc complete equations to deal with. 

 The following separate values must be found from each equa- 

 tion, added together and divided by n. 



%x/n, %x 2 /n, 2^ 3 /n, 



2w/n, SSm/w, 2S 2 w/n, XBhi/n ; 



22 (#, + x n ) +x n+ i—x } 



2 (x Y + x 2 + a? 2 + x % + x n + x n+1 )/ n = 



2(#! + x 2 + x 3 + Xn + x n+ i + x n+2 ) /n = 



32 (#]. 4- x n ) + x n+2 + 2x n +i — x 2 —2x 1 



2 (x v v 2 + x 2 x z + x x x 3 + x n x n+ ! + x n+1 x n ±2 + x n x n+2 )/n = 



2 



32 (a* 2 ) + - 2{# 2 — #i + x z — A 1 }2(«^) + - 2^2—^i ^'3—^1 



