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



In the most simple case of only one element the equations 

 of condition are as follows ; viz. 



e = a x — m 



e 1 = a! x — m! 



e" = a"x — m" 



&c. 

 in which the coefficients of the correction x, viz. a, a', «", &c. 

 are all positive. Now it is manifest that the errors e, e', c", &c. 

 depend upon the correction x ; in so much that when any par- 

 ticular value is assigned to x, all the errors are immediately 

 determined. It appears therefore that there can exist no rea- 

 son for preferring one value of x to another, except the nature 

 of the errors, or the general character impressed upon them 

 on the supposition that the experiments are skilfully executed. 

 When the law of the errors is fulfilled, and when, besides 

 their quantity is confined within the least possible limits, the 

 problem is solved, and we have found that value of the cor- 

 rection which must be preferred to every other. What then 

 is the general character of the errors of a set of experiments 

 made for the purpose of ascertaining the quantity of some 

 physical magnitude, or of approximating to it more nearly 

 than had been done before ? We may suppose at least, that 

 the experiments are liable only to irregular and fortuitous 

 errors; that every cause tending to make the results of obser- 

 vation incline more to one side than to another, has been care- 

 fully investigated and removed ; and in short, that the errors 

 contain no constant part common to them all. What is here 

 said must not be understood exclusively of the case when every 

 error contains a part of the same magnitude affected with the 

 same sign ; the principle evidently extends to all cases when 

 the errors contain parts affected with the same sign although 

 unequal in magnitude, provided these parts are necessarily 

 connected with one another so that they must exist simul- 

 taneously. The point does not turn upon the equality or 

 inequality of the parts, but upon this, That the errors of se- 

 parate experiments must be independent on another and sub- 

 ject to no determinate law. In a repetition of the same ex- 

 periment it is universally the practice to take the arithmetical 

 mean of all the observed quantities, as much more exact than 

 any particular result. Now this rule is founded on the inde- 

 pendence of the experiments and of the errors to which they 

 are liable ; whence it follows that the errors in excess may be 

 expected to balance those in defect ; since no reason can be 

 assigned why the amount of one should be different from the 

 amount of the other. When the quantity to be observed varies 

 from one experiment to another, always containing, however, 



the 



