SECT. 3.] ANY NUMBER OF OBSERVATIONS. 259 



from J = oo to A = -)- oo is ^-, (denoting by TT the semicircumference of 

 the circle the radius of which is unity), our function becomes 



178. 



The function just found cannot, it is true, express rigorously the probabilities 

 of the errors : for since the possible errors are in all cases confined within certain 

 limits, the probability of errors exceeding those limits ought always to be zero. 

 while our formula always gives some value. However, this defect, which every 

 analytical function must, from its nature, labor under, is of no importance in 

 practice, because the value of our function decreases so rapidly, when hJ has 

 acquired a considerable magnitude, that it can safely be considered as vanishing. 

 Besides, the nature of the subject never admits of assigning with absolute rigor 

 the limits of error. 



Finally, the constant h can be considered as the measure of precision of the 

 observations. For if the probability of the error J is supposed to be expressed 

 in any one system of observations by 



and in another system of observations more or less exact by 



h --h h AA 



V/rt 



the expectation, that the error of any observation in the former system is con 

 tained between the limits d and -)- d will be expressed by the integral 



taken from // = &amp;lt;? to // -|- d ; and in the same manner the expectation, that 

 the error of any observation in the latter system does not exceed the limits d 

 and -(- d will be expressed by the integral 



\jn 

 extended from A = d to 4 = -j- d : but both integrals manifestly become 



