202 PROCEEDINGS OF THE AMERICAN ACADEMY. 



essentially diflfereiit forms for these pairs of values of a and /3. Figure 1 

 shows the locus of the equation 



(112) 1/ = —^ [1 - i P3 (s - -i s«)] e'^ 



for p3 = 0, 1, 1.648, 2.2, 3, 5, and Figure 2 the locus of the equation 



(113) y = -^ [I - 1(1 - i ,h) (1 -2 5-^+ 1 s^)] g-'^' 



/X V 2 TT 



for p4 = 0, 1, 1.4, 2, 3. 4, 6, 7, 11. The centre of each figure corre- 

 sponds to s = and the two extremes to s = rt: 4 ; the horizontal line 

 is the axis of abscissie. It will be seen that the ordinates of all the 

 curves are practically for values of s whose absolute magnitudes 

 exceed 4. The curve for which ps = in Figure 1 and that for which 

 P4 = 3 in Figure 2 coincide with the ordinary probability-curve. The 

 curve for which p^ — 3 in Figure 1 and that for which p^ = 7 in Figure 2 

 touch the axis of abscissae ; these values are, by the footnote on page 173, 

 the extreme values of pg and p^, respectively; each figure contains one 

 curve that corresponds to a value beyond these limits, which shows the 

 discontinuity of s or s as the curve passes through the form that has 

 contact with the axis. While p^ is essentially positive, as a mean even 

 power of the relative residual, pg may be negative ; if the sign of the 

 value of p3 that corresponds to either curve in Figure 1 is changed, the 

 curve will be reversed, right and left. In the cases represented in 

 Figure 1, the maximum relative frequency and the probable value of the 

 observed quantity correspond to negative values of s ; in the cases rep- 

 resented in Figure 2, the maximum relative frequency corresponds to 

 s z= for some values of p4, while there are two maxima for the smaller 

 values of p4, the probable value of the observed (juantity corresponding 

 always to s — 0. 



Because s = 1 corresponds to ^ = ^t, that is to f- = p.,, where p^ is 

 the mean square of the residual, it is evident that at least one of the 

 limits s and s must be considerably greater than 1 in absolute magnitude, 

 in any case; but the curves plotted show tliat very little of accuracy is 

 gained by considering values of s beyond ± 4 in the cases represented 

 by them. 



Clark University, Worcester, Mass. 



