ADJUSTMENT OF OBSERVATIONS. 



73 



case shown by figure 26, the group includes 

 six points and the center of gravity is the 

 point indicated by an arrow. The represen- 

 tative line was made to pass through this 

 point and was otherwise adjusted to position 

 bv eye estimate, with consideration of all the ob- 

 servational points and their supposed weights. 

 By this method the observations of greatest 

 weight were enabled to fix one point on the 

 line and were also consulted, along with obser- 

 vations of less weight, as to its direction or 

 attitude. The method obviously left much to 

 personal judgment, but had a rigorous method 

 been attempted it would have been difficult or 

 impossible to avoid the use of nonrigorous 

 judgment in the assignment of weights. 



The line when drawn is a generalized ex- 

 pression, for a single series of observations, 

 of the relation of capacity to slope less a. It 

 is the graphic equivalent, or graph, of a specific 

 equation of the form 



log C= log &! + n log (S-a) . (20) 



which is the logarithmic equivalent of 



<7=6 1 (S-<r) n (10) 



The inclination of the line (tan 6) was next 

 measured, or computed, giving the numerical 

 value of n; and its point of intersection with 

 the axis of log C was determined, giving the 

 value of &,. The values of n, &,, and a for the 

 92 series are shown in Table 15. 



The graph was used, instead of the equiva- 

 lent equation, in computing values of C cor- 

 responding to systems of values of S. The 

 results are given in Table 12 and constitute the 

 data for further generalizations. To compute 

 values of C for a particular observational se- 

 ries, the value of a for the series was first sub- 

 tracted from each value of 8 in the adopted 

 system, then each remainder (S d) was ap- 

 plied as an argument to the graph and the 

 corresponding value of C read off. The range 

 of values thus computed and tabulated was 

 either limited by the range of observational 

 values, or else included a moderate extrapola- 

 tion, which never exceeded 10 per cent of the 

 observational range. 



PRECISION. 



If the position of the straight line in each of 

 the logarithmic plots (fig. 26) had been deter- 

 mined by rigorous methods, it would be pos- 

 sible to compute by rigorous methods the prob- 



able errors of the quantities implied by its 

 position. As only approximate methods were 

 employed in placing them, only approximate 

 measures of precision are attainable, and an 

 elaborate treatment would be unprofitable. 

 The precision. of the attitude of the line, cor- 

 responding to the quantity n, has not been esti- 

 mated; but computation has been made of the 

 precision of its position in the direction of the 

 axis of log C; and also of the precision of the 

 observations, on the assumption that the dis- 

 tances of the observational points from the 

 straight line represent errors of observation. 

 The precision of the position of the line in- 

 volves the precision of the adjusted values of 

 capacity, and also the precision of the coeffi- 

 cient of the equation of adjustment. The 

 method of computation, given below, is also 

 the method employed in various other compu- 

 tations of precision, the results of which appear 

 in later chapters. 



Each logarithmic plot, corresponding to an 

 observational series, was treated separately. 

 The distance of each plotted observational 

 point from the representative straight line, in 

 a direction parallel to the axis of log C, was 

 measured. This distance, interpreted by the 

 scale of the section paper, gives the logarithm 

 of the ratio between an observed capacity and 

 the corresponding adjusted capacity. By using 

 a strip of the section paper as measuring scale, 

 it was possible to read the ratio directly. It 

 was also possible, without computation, and as 

 a simple matter of reading, to subtract unity 

 from the ratio and multiply the remainder by 

 100, thus recording directly the residual, or 

 observational error, as a per cent of the quan- 

 tity measured. The facility of this operation 

 determined the estimation of all errors in per- 

 centage. 



From the residuals thus obtained, probable 

 errors were computed by the following approxi- 

 mate formulas, in which m is the number of the 

 residuals and [v] is the sum of the residuals, 

 irrespective of sign: 



Probable error of an observation = 



0.845 



[v] 



Probable error of adjusted capacities 

 0.845 -M- 



