36 ON THE METHOD OF LEAST SQUARES. 



squares is applicable, are such, Mr Ivory observes, that there 

 exists no bias tending regularly to produce error in one direc- 

 tion, and that the error in one case is supposed to have no 

 influence whatever on the error in any other case. 



From this principle he attempts to show that the method of 

 least squares is the only one which is consistent with the 

 independence of the errors. 



When, however, we speak of the errors as being independent 

 of one another, only this can be meant, that the circumstances 

 under which one observation takes place do not affect the others. 

 In rerum naturd the errors are independent of one another. 

 Nevertheless, with reference to our knowledge they are not so, 

 that is to say, if we know one error we know all, at least 

 in the case in which the equations of condition involve only 

 one unknown quantity, which is that considered by Mr Ivory. 

 For the knowledge of one error would imply the knowledge 

 of the true value of the unknown quantity, and thence that 

 of all the other errors. 



Mr Ivory states the following equations of condition : 

 e ax m 

 e 1 = ax m 

 &c. = &c. 



He thence deduces the following value of x : 



x =<* o + <> 2 an< i those of ee' are 



He remarks that these errors are not independent of one 

 another, as all depend on the single quantity 2ae, which may be 

 eliminated between any two of the last-written equations : but 

 that there is one case in which they are independent of one 

 another, namely, when we assume 2ae = 0, which of course leads 

 to the method of least squares, and that in this case, as we shall 

 have 



