Mr. Ivory on the Method of the Least Squares. 165 



every possible system. It has been shown that the product P 

 has a maximum value determined by the equation 



(_^ . ilUfl) e + (-;' • '-^-^) e'+ &c. = (B) 



Now it is plain that (B) will coincide either with some system 

 S.X<? = 0, the numbers A, x', x" &c. not being all in the same 

 proportion to a, a\ a" &c. ; or it will coincide with the par- 

 ticular system S.ae = 0. The former will be the case when 



— \ — <^_M^il contains <?, and consequently has different values 



for the several errors ; the latter will be the case when the 

 same function is equal to a constant quantity. The equation 

 (B), in which are contained all the most probable systems, 

 cannot coincide with any case S. X^ = 0; for then, according 

 to the demonstration of Laplace, the probability would be 

 less than in the system S.fl^ = 0, v.hich is absurd. There- 

 fore the only supposition that can possibly be true is the iden- 

 tity of (B) with '$>.ae =0. But this requires that 

 1 d-j (£^) _ _ ;^2 . 



and consequently, ^^^s) _ ]cc~''*^'. 



Since the equation S.«^ — 0, makes the function S.e' a 

 minimum, it follows incontestably, from the demonstration of 

 Laplace, that the minimum of S.f' must coincide with the 

 maximum of the product P ; which is the same condition to 

 which we have already brought the determination of the pro- 

 bability of an error by a different mode of reasoning. There- 

 fore the investigation of Laplace, whatever merit it may have 

 in other respects, is neither more nor less general than the 

 other solutions of the problem. 



The analysis of Laplace is different from that of other ma- 

 thematicians in reversing the order of investigation. It has 

 been most usual to begin with seeking the .law of the proba- 

 bility of an error ; and, when this is found, the chance of a 

 given combination of the errors is derived from it. Laplace 

 begins with computing the chance of a given combination of 

 the errors by means of the doctrine of combinations ; but it 

 is manifest that the result obtained, when compared with the 

 equations supposed to subsist between the errors, leads to a 

 particular law of probability. ITpon the premises laid down, 

 both methods lead to the same conclusions, and the demon- 

 strations obtained in both ways are eciually extensive. The 

 investigation of Laplace is more analytical an(l more philoso- 

 phical, as it requires no jjrevious discussion of the law of the 

 probability of an error; but, on the other iumd, it is confined 



