198 PROCEEDINGS OF THE AMERICAN ACADEMY. 



the same absolute magnitudes are ecjual, that is, positive and negative 

 residuals are equally probable. 



The calculation of the true values of the C's by (80). (92), and (93) 

 will be facilitated by tables of the values of S,„ /?„,_i and of the integral 

 (94). The values of S,„ R„,^i will be re<iuired for such values of m as 

 are suffixes of the P's that are not negligible, which will Ije small, un- 

 doubtedly, in all actual cases ; tables of S„^ R,„_i for m = 3, 4, 5, G will 

 probably be sufficient. For the values of the integral (94) the ordinary 

 tables of 



2 / 

 (104) <D (y) = —= I e-*' • dt 







can be used ; namely, 



y 







so that, observing that <J> (— y) = — <I> (y), we have 

 (105)/e--* = /i[*(^)-*(^)] 



where s and — s are both positive. 



XII. Probable Values. 



We have defined the probable value of the observed quantity to be that 

 value that is just as likely to be exceeded as not ; if the probable residual 

 ^, and the probable relative residual s^ be taken to correspond to this 

 probable value z^,, ^^, and s^, will be determined by the equation 



(106) |V (0 ■ d$ = ff(s) -ds^^ 



i }. 



and Zp by 



(107) Zp = Co + ^,„ 



in accordance with (3), so that the arithmetical mean is (in general) not 

 the probable value even of a quantity directly observed (as it is assumed 

 to be in the method of Gauss). 



