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



than any one of the former ones up to the sum of the 6th 

 powers. AppUed to the above example, r becomes 



r = 5"''914 + 0"'-864, 

 or the limits 



5"'-050 and 6"'-778. 



In the demonstrations hitherto given, it has frequently been 

 necessary to conclude from the probability of one value to that 

 of another value depending on that of the first in a simple man- 

 ner. For the sequel it becomes necessary to solve the general 

 problem. If we know the most probable values of certain inde- 

 pendent magnitudes x, <r', x", &c., and the different limits 

 within which these most probable values will lie, if any deter- 

 minate probability is to be ascribed to them, to determine the 

 most probable value of any function of these variables, 



X =f{x,os',os", . . .), 

 and also the limits within which X has the same determinate 

 probability. As, when we know the value of r in a magnitude de- 

 duced by observations, we can at once find h, e,, and all the other 

 functions of the errors, as well as their complete law ^ (A), the 

 problem may be proposed in this way : for x, a}, ^', the most 

 probable values a, a', a" having been found independently of 

 each other with the probable errors r, r', r" . . . ., it is required 

 to determine the most probable value of X =f{x, a?', a^'. . .) and 

 its probable error. 



To begin with the simplest case, let X be a linear function of 

 one unknown quantity 



X = a<r. 

 In all the cases in which ^ = a, X = a «, consequently this 

 wiU also be the most probable value of X. So also the cases 

 in which .x- lies between a—r and a + r are equal in number 

 to the cases in which X lies between a.a~ a.r, and ua -{- ur; 

 or 



'K =z a.a + ctr, 

 where the last member denotes the probable error of X. 



Now, in the second place, let X be the simple linear function 

 of two variables 



X = a? + a?'. 



For the sake of more convenient expression, let us now intro- 

 duce, in lieu of the probable error, the weight of the values a and 

 a'. If an observation, of which the probable error is w, be taken 



