Mr. Ivory on the Method of the Least Squares. 7 



will, therefore, fix upon the best mode of solution by the same 

 rule that we employ for ascertaining the most advantageous 

 of several sets of observations. That mode of solution is, 

 therefore, to be preferred, in which the mean of the squares of 

 the errors is the least. But, in all possible ways of solving a 

 system of equations of condition, the number of the errors is 

 constantly the same ; and therefore the mean quantities are 

 proportional to the total sums. And hence we must conclude, 

 as we have already found by a different train of reasoning, 

 that the most advantageous result will be obtained when the 

 equations ^re combined so as to render the sum of the squares 

 of the errors a minimum. 



It must, however, be allowed that every thing which has 

 just been said of the sum of the squares of the errors, and of 

 the mean of the squares, will equally apply to the sums and 

 the mean quantities of any of their even powers. Thus the 

 latter demonstration leaves it doubtful whether it is the sum 

 of the squares of the errors, or the sum of any other of their 

 even powers, that must be a minimum in order to obtain the 

 most advantageous result. But it is easy to exclude the other 

 even powers, and to render the demonstration absolute with 

 regard to the squares. The equation of the minimum of the 

 function, 



is this, 



de ^2n-l d e' ,2^-1 , d e" „2n-\ 



1^' +Vr' + J7' +&C. = 0, 



or, 



a^2;i-l + q'^'2«-1 4. a^ e"'2n-t + &c. = 0: 



and as this is true in alj relations of a, a', a", &c., it will be 

 true when all these coefficients are equal; in which case the 

 equation will become, 



e 2/1-1 4. e' "211-1 4. ^'2/1-1 4. gj^c. = 0. 

 But the supposition of the equality of a, a', a", &c. places all 

 the errors in like circumstances ; and it is universally admitted 

 that in a series of observations made in like circumstances, 

 the simple sum of the errors is equal to zero, and not the siun 

 of their cubes, or fifth powers, or any other of their odd 

 ■powers. By this argument the demonstration is restricted to 

 the squares of the errors, and the other even powers are ex- 

 cluded. 



The first of the two demonstrations founded on the prin- 

 ciple suggested by Cotes is clear and simple, and leads ex- 

 clusively to the method of the least squares. It is probably 

 the best and most solid demonstration that can be given. 



The 



