334 J. F. ENCKE ON THE METHOD OP LEAST SQUARES. 



It follows from this formula, that the probability that an 

 error lies between A and A + d A,is 



= A,-»-v^, (7.) 



and the probability that it lies between the arbitrary limits a 

 and b, 



a/ Try A = a 



dA. (8.) 



Calling the number of the errors unity, this integral expresses 

 also the number of errors which should occur between a and b 

 according to the law, and which will occur very approximately 

 if the number is sufficiently great. If we make 



h A = t, 

 the integral takes the form 



\/7ry t = ah 



If we take for the limits an equal positive and negative value 

 — ah\.o + a h, then on account of the even power of t in the 

 differential, we may write 



e~''dt; 

 t= 



and we may thence, by means of a table giving this integral 

 for successive values of a h, obtain a clear representation of 

 the distribution of the errors, without regard to signs, but 

 simply in respect of their magnitudes, proceeding from to the 

 extreme limits. Such a table is appended (Tab. I.). It is de- 

 duced directly from the table for the integral / e~^ d t in 



BessePs Fundamenta Astronomies. The calculation of such a 

 table from the developement of the integral according to as- 

 cending and descending powers of t, or according to a continued 

 fraction, is found frequently given, as this remarkable function is 

 applied in many ways in diiferent researches. 



This table shows at the same time, how rapidly the number 

 of errors included in equal intervals of the value of t decreases 

 in the higher values. It justifies, therefore, our assumption 

 of the limits — oo and + co in lieu of the actual limits, which 

 must be narrower, although they are not susceptible of being 



