4 Mr. Ivory oti the Method of the Least Squares. 



O + e =V. Wherefore, by substituting the development of 

 V, and putting ?k = — V, we shall obtain this eq^iation of 

 condition, 



e = — m + a X + bi/ + cz + &c. 



Every observation will furnish a like equation. In this pro- 

 blem, therefore, we have a system consisting of any number 

 of equations, viz. 



e = — m -'r ax ■\- b y -\-c z -\- &c. 



e' = — m' + a'x + b'y +c'z +&c. 



e"= — 7ti[+ a''x + Wy +c"z +&c. 



&c. 

 And it is required to determine the corrections x, y, z, so 

 as to make the errors e, e', e", &c. either absolutely equal to 

 zero when this is possible, otherwise so that they shall be 

 contained within the least limits. When the number of the 

 equations is just equal to the unknown quantities x, y, z, &c., 

 the errors being supposed evanescent, the problem will come 

 under the usual rules of algebra, and will admit of an exact 

 solution. But in the cases that occur in practice, the num- 

 ber of equations being greater than the corrections to be 

 found, no mode of solution will entirely annihilate the errors, 

 and we must be content with reducing them to the least pos- 

 sible quantities. 



Cotes appears to have introduced the use of equations of 

 condition. In the most simple case of only one element, the 

 system of equations is as follows, viz. 

 e = a X — 7n 

 e' = a' X — 7u' 

 e" = a"x — ?«" 

 &c. 

 Supposing every error equal to zero. Cotes finds the several 



values of X, viz. , -, , — ^, &c.; and, by applying these cor- 

 rections, as many determinations of the element are obtained 

 as there are observations. Now, let all the values thus found 

 be set off, on the same side, from a given point in a straight 

 line ; and let the weights a, a', a", &c. be appended to the 

 extremities of the several parts ; then the distance of the cen- 

 tre of gravity of all the weights from the given point will, 

 according to Cotes, be the most advantageous value of the 

 element. By this process the correction to be added to the 

 appi'oximate element conies out equal to 



m -{- m' -\- m" -\- &c. 

 a + n' -(- a" -\- &c. ' 



We shall obtain the result of Cotes's method more simply 

 by adding all the equations of condition, and making the sum 



of 



