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



and it is proportional to the probability of ». To find the 

 fraction equal to the absolute chance, we must divide the fore- 

 going expression by the sum of all the probabilities for every 

 possible value of u, that is, by the value of the integral 



(S.Xn)'2 



--^==Jduc S^^ 



taken between the limits + oo. The integral is equal to 

 -r ; and hence the absolute probability of u is equal to 



(S.Xa,^ 



a/^S.x" 

 Now if we put ti = 0, the value of the above expression is 



— ;- X — '-— i which is therefore the probability that the ele- 

 i/V -v/s.x^ 



ment x is equal to ^ , or that the errors of observation coin- 

 cide with the system S.X^ = 0. The factors K, \', x"&c. must 

 next be determined so as to make the probability a maximum, 



or so as to make ^ a minimum, which is the conclusion 



S . Xa 



Laplace has arrived at, p. 319, by considering the matter a 

 little differently. It is found that the maximum will take place 



when i. _ _^' _ ^ ^p . 



o a' a" 



and consequently the most probable system of errors is found by 

 the equation S.ae = 0, which is the condition that makes S.^ 

 a minimum. It is to be observed too, that no step of the in- 

 vestigation depends upon the value of h, or upon the law of 

 probability, which is left indeterminate. 



The foregoing reasoning is detailed particularly in order 

 to prove clearly, what might have been inferred from the ana- 

 lysis of Laplace, that the system of errors S.ae = 0, is more 

 probable than any other system S. A^ = 0, in which X, a', a" 

 &c. are not all in the same proportion to a, a', a" &c. Now 

 this being proved, I say, that the law of the probability of an 

 error is thereby fixed, and no longer remains indeterminate, 

 as stated hy Laplace. Let <p(<?0 denote the probability of the 

 error e : then the probability of any system, or the probabi- 

 lity of the simultaneous existence of the errors e, c\ e" 6^c. of 

 that system, is equal to 



-1-X ^(c').<p(/^).?(/'^)cScC. = ^, 



the letter H representing the sum of all the values of P for 



every 



