﻿IN THE METHOD OF LEAST SQUARES. 201 



The limits of effective accuracy appropriate to the numerical representation of the 



coefficients a,h may be investigated in the following manner : — 



If we determine cV, , ^, by the method of least squares from the equations, 



(38.) (a — da) Xi-\- {h — dh) tji -\- -|- '" — (Z /n = Ci, weight = w, 



we shall have fi-om (8) and (38), i£ da, db , which may be employed to represent 



the errors of the adopted coefficients, are small, 



(39.) 



P {xi — x) -\- P' (yi — !/) -Jf- = aw (— e-\-xda-\- ydh-\- + d m j + 



K // is the probable value of d m »t/w, the most suitable values oi x da, y dh . . 

 evidently fulfil the conditions 



(40.) xdaS^w—ydhy/w = dm\^w = ii', xda'Vw'^ydb'Vw' = d m' >/ w' = ii' , . 



Observing that we may substitute in the second members of (39) the probable values 



e <i a X, e' <^ a' X , e < by, e' < b' y , w^e may conclude, by comparing (39) 



with (11), (12), and (6), that, if / is less than the limit 



V« + 2 

 n' denoting the number of indeterminates in (38), the difference fj — e of the probable 

 errors of Xi, y, derived from (38) and oi x, y derived from (8) will be 



(42.) f, -£<ig'^£. 



Consequently, if / < ^ , s^ — s will be less than —^ s or less than in 11. 



When / is known, the limits for admissible values of da, db in (38) will be 



(43.) da<^ — ^^ , da' <^ — ''^ dh=^'- da, db' — -da' d m =z x d a, dm' = xda' 



X \^w X \^w' y y 



K the mean values of ax, by , irrespective of their signs, are all of the same 



order of magnitude, we may substitute in (43) the a priori probable values 



(44.) .<?! I " ■ ^ = ?, 



where n is the number of equations (38), n the number of indeterminates, and, taking 

 the means without regard to signs, 



(45.) A = mean value of a VuT, B = mean value of h \/w M = mean value of m ^/w . 



