ON THE METHOD OF LEAST SQUARES. 37 



each error is determined by " the quantities of its own experi- 

 ment." But this reasoning is perfectly inconclusive. In the 

 case supposed, ee &c. are as much connected together as in 

 any other, as may be shown by eliminating 2am between the 

 equations 



and besides, apart from any mathematical reasoning, it is clear 

 that as if we know one error we know all, so also if we assign 

 any value to one, we have in effect assigned values to all, whe- 

 ther we use the method of least squares or any other. 



Moreover, e is not determined by the quantities of its own 

 experiment alone, since Sara involves the results of all the ex- 

 periments ; there is no difference between this and the general 

 case, except that ^ae has ceased to appear in the equations. But 

 suppose we multiplied the equations of condition by any func- 

 tion of a, we might deduce the following values of x and e : 



. m 



~~ 



e = m + 



5' = -W + ^ 



a . (f>a 



Mr Ivory's reasoning would apply word for word as before, 

 and would show that the best mode of combining the equations 

 of condition was to employ the factors </>a, (pa, &c. whatever be 

 the form of </>. As it thus would serve to establish, at least 

 apparently, an infinity of contradictory results, the inference is 

 that in no case has it any validity. 



I have now completed, though in an imperfect manner, the 

 design indicated at the outset of this paper, namely, to give 

 an account of the different modes in which the subject has been 

 treated, and to simplify the analytical investigations. If I have 

 succeeded in doing this, the present communication may tend 

 to make a very curious subject more accessible than it has 

 hitherto been. 



