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



to avoid wholly the greater constant errors, — or to lessen them as 

 much as possible, — or to bring their influence so far within the 

 power of computation, that the remaining constant errors in one 

 mode of making the observations may appertain to those sources 

 of error which in other modes of observing can exist only in a 

 different degree. In this case, it is as important to multiply 

 the methods themselves as the observations in each ; and by 

 making as many repetitions as possible, and by varying the 

 methods themselves as much as possible, the nearest approach 

 is made to the truth. This distinction between constant and ir- 

 regular errors does not influence the application of the calculus 

 of probabilities, so long as we do not know whether any and what 

 constant errors exist. Their existence may be ascertained, if, on 

 comparing together the results of different methods, we find 

 that a greater difference exists between them, than the treatment 

 of the observations by each method separately would justify 

 us in expecting. For the most part, the midtiplication of the 

 observations according to one method is easier to obtain, and is 

 more frequently met with, than the multiplication of the methods 

 themselves. On this account the result deduced as most proba- 

 ble is usually a partial one ; and, in order to come as near as 

 possible to the pure truth, the chief object of attention should 

 be to avoid every possible constant error. In the sequel 

 this distinction will be disregarded ; it only causes the estima- 

 tion of the exactness of such a partial result to be always 

 somewhat faulty, — a circumstance so much the less influential on 

 the general consideration, as the estimation itself lays no claim 

 to absolute certainty. 



If now, with the following conditions, A a maximum for 

 A = 0, ^ A an even function of A, and ^ A = for A > a, — 

 we combine the remark drawn from experience, that in general 

 smaller errors are more frequent than greater ones, — that in 

 approaching a, the extreme limit, the number of errors de- 

 creases with great rapidity, — and that between A = and 

 A = a there is in general no value of A for which (^ A is im- 

 possible, or that all errors from to « may exist, — then the 

 march of the function may be assigned a priori. A geome- 

 tric consideration may be here employed to facilitate the con- 

 ception of it. If the values of A be taken as abscissae, and 

 the <f> A belonging to them as rectangular ordinates, the curve 

 of probabilities on both sides of the axis of ordinates will be 



