Mr. Ivory on the Method of the Least Squares. 5 



of the errors equal to zero; for the vahie of x found in this 

 manner will coincide with the expression just set down. And 

 hence we may conclude, that the mode of combining the 

 equations of condition imagined by Cotes is not the most ad- 

 vantao-eous. For, as the total sum of the errors is equal to 

 zero, an error of a given amount in any of the equations has 

 the same influence on the value of the correction. But it is 

 very evident that the error e produces a small variation in 

 the value of x when a is a great number ; and, on the con- 

 trary, a great variation when the same coeflicient is inconsi- 

 derable. The procedure of Cotes is therefore just and unex- 

 ceptionable only when all the coefficients, «, a', a", &c. are 

 equal, or when all the errors are in like circumstances. 



Reflecting on what has just been said, we may, by a slight 

 alteration in the procedure of Cotes, deduce a better way of 

 combining the equations, and one which is more advantageous- 

 than any other. The influence of the error e on the value of 

 the correction x is less when the coefficient a is greater, and 

 it increases when the same coefficient is diminished in magni- 

 tude. This is exactly similar to a lever which is to pioduce 

 a given effect; for the lever must be shortened when the sus- 

 pended weight is greater, and lengthened when the same 

 weight is less. Draw a straight line, and from a given point 

 in it set off" all the positive errors on one side, and the nega- 

 tive errors on the other side; then, having suspended the 

 weights a, a\ a", &c. from the levers e, e', e", &c., make the 

 levers in equilibrio about the common fulcrum. By this con- 

 struction the errors will be so determined that the coefficients 

 a, a', a", &c. will have each its proper influence on the value 

 of the correction x. The equation of the equilibrium is 



ea + e'a' + e"a" + 8ic. = 0; (A)* 



and, if we observe that x is the only variable quantity in the 

 expression of the errors, the same equation may be thus writ- 

 ten, viz. 



e^±+e'^ + ^''-f +&C. =0, 



dx dx dx 



which determines the minimum of the function 



^ + ^'2 + e"= + &C. 



• It is evident that here, as well as in what follows, we speak only of 

 irreciilar and fortuitous errors that have no constant part. If we suppose 

 a part common to all the errors, there would not he an eciuii.bruini of the 

 levers but a preponderance. Let + /denote the constant error; then. 

 in place of eciuat. (A), we should have this which follows, viz. 



« e + a c + «" f" + &c. = 4 (a + «• + «' + <!vc. X /; 

 and/ would he the distance hetweeu the common fulciuni of all the levers 

 and the centre of gravity of the weights a, a, a', &c. 



I hus 



