336 J. F. EXCKE ON THE METHOD OF LE.VST SQUARES. 



isolated observation does not exceed r, supposing the value of 

 r for the class of observation to be known. 



On account of its frequent use, the value of the integral 



-/- / e dt has been also given in a second table, arranged 

 according to an argument in which the value of r has been as- 

 sumed as unity. This table gives for the argument — the 

 value of f = ^~ 





so that it shows immediately how many errors will occur up to 

 a determinate error A (always without reference to the sign), 

 provided the proportion of the given A to the probable error 

 be known. It further facilitates a view of the distribution of the 

 errors according to their magnitude. If half the number of all 

 the errors are less than an error =r, then among 1000 observed 

 errors, there will be 823 less than 2 r, 957 less than 3 r, and 993 

 less than 4 r. There will not be more than one error greater 

 than 5 r. 



By means of this view of the probable error, we may also ob- 

 tain a clearer view of the signification of the constant h. In 

 dilFerent kinds or sets of observations the en'ors always follow the 

 same law, which is expressed by i^ A. The difference of any 

 one kind or set from all others depends, therefore, solely on the 

 value of the constants h, and these again afford the means 

 of comparing together observations of different kinds in re- 

 spect to exactness, and thus enabling them to be subsequently 

 combined. 



With two kinds of observations, one of which has the constant 



h and the other the constant /<', the integral f ^ A d A, taken 



up to any assigned limit, will have equal values, if the value of 

 the limit, determined in both cases by the variable t, is the 

 same. Or (as in one t = h A, and in the other t = h' A', the 

 errors of the second kind being designated by A') there will be 

 as many errors in proportion to the whole number in both kinds 

 within the limits A and A', if we determine one value from the 

 other by the equation 



h A =h' A'; (11.) 



