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



The equation (A) must be considered as a generalization, 

 drawn from the nature of the equations of condition, of the 

 rule universally admitted by astronomers, that the sum of the 

 errors of a numerous set of observations made in hke circum- 

 stances is equal to zero ; which rule is itself contained in the 

 equation, being that particular case of it when the coefficients 

 are all equal. But although the first demonstration be suffi- 

 cient to establish the proposition, yet the second, as it throws 

 additional light on the matter, may not be deemed super- 

 fluous. 



It is easy to apply to two or more elements what has been 

 proved with respect to one. In the case of two elements we 

 have this system of equations, viz. 



ez=.ax + by — VI 



e' = a' X + b'y — m' 



e"=a"x-\-V<y-m" ' 



&c. 

 The corrections x and y being independent of one another, 

 the errors must be determined so as to give their proper in- 

 fluence both to the coefficients a, a\ a", &c. and to the co- 

 efficients b, b', b", &c. The levers e, e', e", &c. must there- 

 fore be in equilibrio when the weights a, a', a", &c. are sus- 

 pended ; and again, when these are removed, and the other 

 weights b, b', b", &c. are substituted for them. These two 

 slates of equilibrium lead to the equations 



ae + a'e + a" e" + &c. = 



be + b'e' + b" e" + &c. = 0, 

 which are sufficient for finding both the unknown quantities 

 X and y. The same two equations may be otherwise written 



thus, viz. ^^ ^^, de" „ . 



.' e + -J- e' + -— e' + &c. = 



ax ax ax 



<ie , de' „ , de' „ o ^ 



--- e + -r- e" + — - e" + &c. = 0, 



dy dy dy ' 



of which the first determines the minimum of the function, 



^ + e"' + e" = + &c. 

 relatively to the variable x\ and the other determines the 

 same minimum relatively to the variable y. Both the equa- 

 tions together determine the absolute minimum of the func- 

 tion relatively to both the variables ; which condition there- 

 fore contains the full solution of the problem. 

 In the case of three corrections, viz. 



e = a X + b y + c z —m 



e' = a' X + b'y + c' z —m' 



^" = a!Kx + Wy -f c'z —m" 



&c. we 



