﻿210 



ON THE USE OF EQUIVALENT FACTORS 



Coef. of 

 ilL. 



— _ 371.03 + 5.97864 + 



= — 112.76 + 



= — 1108.40 



= + 274.45 



= + 117.10 



= + 30.76 



III. 



Coef. of Coef. of Coef. of 



rf)-. (1.71. cltf. 



7296.25 — 0.06706 — 2.33932 — 



2048.68 + 0.02871 — 0.44288 + 



+ 7.18029+ 11.56325 + 



+ 9.07933 — 



+ 



Giving, for the unknown quantities, 



I. (Revised solution.) 



di = — 7.76 

 da =— 51.76 

 (46.) d(p =— 33.20 



d,r = +219.19 

 dy = -\- 0.05401 

 dL = — 15.68 



II. 



— 3.77 



— 46.94 



— 22.41 

 + 186.17 



+ 0.08963 



— 56.32 



III. 



— 5'.'76 



— 53.85 



— 33.50 

 + 212.41 



— 0.05115 



— 14.58 



These results will form the basis for a comparison of the relative probability of the 

 three systems. 



In view of the apparently gross discrepancies of the solutions II. and III. from I., 

 and with the admitted fact before us that the latter reduces perfectly to a minimum the 

 sum of the squares of the residual errors, we should at first sight scarcely be disposed 

 to admit that there is in reality no sensible diflFerence in the accuracy of the three 

 systems. This luifavorable impression must have its origin in the erroneous infer- 

 ence, that, inasmuch as I. is theoretically the best of all solutions, the errors of the 

 results of other systems must be proportional to their deviation from 1. 



For instance, in I. we have dL^ — 16", but in II. d L =: — 56"; from which 

 is inferred that in II. the error of d L is proportional, or nearly so, to — 16" -|- 56" 

 = -|- 40" ; and in a similar way, d fl, d cp, Sec. wiU be condemned or approved by 

 the standard of their disagreement or accordance with the corresponding values 

 in I. 



Such a mistake may, we repeat, be easily committed, in a cursory glance at the num- 

 bers in question. Tliat it is a violation of the fundamental principles of the method 

 of least squares will appear on a very little consideration. 



To apply the criterion proposed in that method for determining the relative proba- 



