ADJUSTMENT OF OBSERVATIONS. 



61 



particular series of observations but from 

 several related series. It was assumed to be a 

 typical or representative curve for the func- 

 tion log ^=/i(log 8); and the corresponding 



FIOUBE 19. Logarithmic graph of C=/(S), for grade (Q), M 0.66 foot, 

 Q=0.734 ft./sec. 



plot on ordinary section paper (the line BD in 

 fig. 20) was assumed to represent typically 

 the function C=f(S). The coordinates of the 

 line BD are given in Table 5. 



TABLE 5. Values of capacity for traction, graphically gen- 

 eralized from data of Table 4 (G), for w=0.66 foot and 

 Q0.734ft. 3 lsec.; corresponding to the curve of log C= 

 /! (log S) in figure 19, and the curve BD in figure 20. 



A number of tentative formulas were now 

 compared with this empiric line, their param- 

 eters being computed so that they would fit, as 

 nearly as practicable, the values of C in Table 

 5. Certain functions, including the simpler 

 functions of the circular arc and the exponen- 

 tial function <7=e"< s + a >, could not be fitted, 

 even approximately, to the data; but the fol- 



lowing functions yielded curves closely re- 

 sembling that of figure 20: 



(6) 



(7) 



(8) 



(9) 



.(10) 



(11) 



Functions (6), (7), and (8) are special cases 

 of the general formula of interpolation with 

 integral exponents : 



No. (11) is a somewhat involved power 

 function suggested by results of a preliminary 

 discussion of the laboratory data. Nos. (9) 

 and (10) are special cases of the general para- 

 bolic function 



(z + a)" = 



.(13) 



and have the virtue of facilitating the graphic 

 treatment of the material. Their logarithmic 

 equivalents are, respectively, 



log (<7+/l)= log b + n log S _______ (14) 



log C"=log b + n log (S-a) _______ (15) 



and, as each of these is the equation of a 

 straight line, the graphic derivation of the 

 exponent, by means of logarithmic section 

 paper, becomes a simple matter after the 

 value of ^ or a has been determined. 



The adjustment of equations (6) to (11) to 

 the specific data in Table 5 gives them the 

 following forms, (6a) being derived from 

 (6), etc.: 



C= -29.55 + 67.59^-7. 194S 3 ______ (6a) 



C -- 12.865 + 44.08S' 2 ______________ (7a) 



<7= - 19.25 + 16.945 + 34.48S 2 _______ (8a) 



(7=-10.0+41.2S"- M _______________ (9a) 



0-70.5(5 -0.39) 1 *-. . (lOa) 



(7-31.2* 



2.68 

 



-(Ha) 



When the curves corresponding to these 

 equations are plotted for the region covered 

 by the empiric line BD, they coincide very 

 closely with that line. The greatest departure 



