168 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



A, + A,-}-A,i-... + A„ = [A], 



as is customary in the theory of errors. Let Zq be the arithmetical mean 

 of the observed values, so that 



(0 



n 



Let X be the error of an observation in general, Xi, X2, x^, . . . , x„ the 

 errors of the n given observations, and Xq the arithmetical mean of these 

 errors, so that 



fx — z — Z,Xi = zi — Z, X2— Z2 — Z, X3 = Z3 — Z, . . . , x„ = z„ — Z, 



(2) 



n n 



Z — Zq — Z ; 



therefore, the arithmetical mean of the errors is the error of the arithmet- 

 ical mean of the observations. Let $ be the residual of an observation 

 in general, — that is, the excess of the observed value over the arithmet- 

 ical mean of the given observed values, — and let $1, i^t ts, • • • , ^n be 

 the residuals of the several observations, so that 



(3) 



Zq = (X ^ Z) — (X^, + Z) = X — Xf^; 



therefore, the residual of an observed value is the residual of its error ; 

 we may call it simply the residual of the observation. Let p;. be the 

 mean ^--th power of the residual, so that 



(4) 



in particular, 



(5) 



Pk 



'po = I in 



m ; 



[1] 



Pi 



[^^] - - [^-] 



0. 



The positive square root of the mean square of the residual shall be 

 called the mean residual (it is called the mean error in tlie ordinary 

 theory) and shall be represented by jx, so that 



(6) /A = \/;>2 or p^ = fi^ 



The ratio of the residual to the mean residual shall be called the relative 

 residual and be represented by s, while the ratio of the mean ^-th power 

 of the residual to the k-ih power of the mean residual shall be called the 

 relative inean ^-th power of the residual and be represented by p^, so that 



