206 Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 



causes had disappeared from a given series of observations by assigning different forms to /, 

 and comparing the different values thus found for a. 



No satisfactory reason can be assigned why, setting aside mere convenience, the rule of the 

 arithmetical mean should be singled out from the other rules which are included in the general 

 equation 2/ (a? - o) = 0. 



Let us enquire, therefore, whether there is any sufficient reason for saying that the rule of 

 the arithmetical mean gives the most probable value of the unknown magnitude. In the first 

 place, it is only one rule out of many among which it has no prerogative but that of being 

 in practice more convenient than any other : in the second place, if this were not so, it would 

 not follow that in the accurate sense of the words it gave the most probable result. This 

 objection I shall defer for a moment, and proceed to consider the manner in which Gauss makes 

 use of the postulate on wliich his method is founded. 



From the first principles of what is called the tlieory of probabilities a posteriori, it appears 

 that the most probable value which can be assigned to tiie magnitude whicli our observations are 

 intended to determine, is that which shall make the a priori probability of the observed phe- 

 nomena a maximum. That is to say, if a be the true value sought, «, being the value observed 

 at the first observation, le-i the corresponding quantity for the second, and so on, the errors at 

 the first, second, &c. observation must be x^- a, Xo- a, kc, respectively; and if (pe.di be the 

 probability of an error e in any observation of the series, the quantity whicli is to be made a 

 maximum for a is proportional to 



(f> ('^1 - a)<p{x2~ a)... <p {x,- a). 



Equating to zero the differential of this with respect to a, we find 



(iBi - a) (p {x„ - a) 



as the e(|uation for determining a in x. Let ^ = \|/, then it becomes 



2" >|/ («! - a) = 0. 

 Now we have assumed that the most probable value of a is given by the equation 



2J (a - a) = : 

 and it is impossible to make these equations generally coincident, without assuming that 



\j/e = me, m being any constant; 



(p'e 



hence ~ — = me, 



<pe 



and (pe = Ce^-'"'\ 

 Now as the error 6 is necessarily included in the limits - eo + eo , we must have 



/■ 



d) e d e = 7= 1> 





or if we adopt the usual notation, and replace m by Zh", 



„ h h 



C = —y- , and (he = — y 



e 



