Mr. Ivory on the Method of the Lead Squares. 87; 



and as the integrations produce constant quantities only, the 

 same probability will be equal to the expression, 



The constant k will be determined by observing that the in- 

 tegral 



kj due J 



taken between the limits ±co, comprehends every possibje 

 error, and it must therefore be equal to unit. Hence — ^-p— 

 = 1 ; and k = "^.. Wherefore, finally, the probability of 



the error u, in the value A. found by the method of the least 

 squares, is equal to 



^' 

 If we compare this expression with the error of an ongmal 

 observation, it will appear that the precision of A, the value 

 of the element found by the method of the least squares, is 

 to the precision of the actual observations, as h P to //, or as 

 P to 1. The probability that the true value of the element 

 is between the limits A(l + S) is equal to the integral 



— —J du c 

 •J ■* 

 taken between the limits + A 8. 



It is easy to transfer what has been proved with respect to 

 u the error of A, to v and w, the errors of B and C. As the 

 solution we have given extends to three elements, it will ne- 

 cessarily comprehend the subordinate cases of one and two 

 elements ; and there is no difficulty, except the length of the 

 operations, of applying the same analysis to any number of 



It is not my intention to treat of the practical details of 

 this Theory, but merely to lay before the reader that particu- 

 lar view of its j^rinciples which appears most natural and phi- 

 losophical. All that part connected with the doctrine of 

 chance, is founded on the hypothesis that in all cases the pro- 

 bability of an error depends precisely in the same way on the 

 mamiitude of the error, or that it is always the same iunction 

 of the error. Now, I believe, it will be allowed that the 

 crounds of this supposition are much less^ sure than the evi- 

 dence adduced in i)roof of the method of the least squares. 

 There would therefore be a great logical fault ni maknig the 

 most advantageous solution of a system of ecjuations of con- 



dition 



