J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 321 



symmetrical. A maximum maximorum will be found at A = 0. 

 From this point forward, according to the law, the curve will be 

 drawn continuously, so that in the neighbourhood of A = «, 

 it will approach the axis of abscissae very rapidly. Hence 

 follows another circumstance of great importance in the se- 

 quel. The absolute limit a can never be strictly determined : 

 but as in the neighbourhood of a the ordinates ^ A decrease 

 very rapidly, we may without any sensible error assume the 

 limits — CO and + oo , instead of the values of a, provided 

 the function, which within the values and a should agree 

 with the march of the curve, has the property of decreasing 

 constantly as A increases. For in the rapid approach to the 

 axis of the abscissae, so soon as A approaches a, each func- 

 tion which beyond a decreases still more, and was before ap- 

 proaching rapidly, will give for its values between + a and + oo 

 only insensible magnitudes. 



The definition of ^ A implies, that when the number of ob- 

 servations is so great that all errors will occur, each in due pro- 

 portion of frequency, 



w <^ A + m A' + w </) A" . . . . = m, 



+ ^ 

 or S (<^ A) = 1. 



— 00 



Hence we perceive that if the number of A be infinite, M'hen 

 all the gradations from A = to A = a are taken into account, 

 the function (j> A will be infinitely small for any given error A. 

 We may express this condition more conveniently, in the lan- 

 guage of analysis, by not considering the probability of one 

 determinate error only, but the probabihty of all the errors 

 lying between the infinitely near limits A and A + rf A. Within 

 these infinitely near Umits, the value of <^ A may be regarded as 

 constant. Hence the probability of the errors between A and 

 A + c?Ais^Ac?A; and the probability of the errors between 

 the limits a and b is equal to the sum of these elements within 

 the given limits, or 



^ArfA. (1.) 



For the limits — x and + so , which include all errors, it becomes 



/+00 

 <;!. A^/A = 1, (2.) 



