Mr. Ivory on the Method of the Least Squares. 85 



of n when the number of the observations is great, it follows 



that — will be greater or less according as the errors, taken 



upon the whole, are more or less considerable ; that is, ac- 

 cording as the observations are less or more exact. 



4. Let lis now consider this system of equations of condi- 

 tion, viz. 



e ■= a X -{■ by + c z — m 

 e' = a' X + V y -i- c' z — m' 

 e"= a"x + Wy + c"z — w" 

 &c. 

 the quantities x, y, 2, as well as e, e', e", &c., being indeter- 

 minate. Put e, g', e'', &c. for the particular values of ^, e', /',&c. 

 in the most advantageous solution, or when the sum of the 

 squares is a minimum ; and let A, B, C be the corresponding 

 values of x, y, s, : we shall have 



e=aA + iB + cC — OT 

 g' = a' A + A' B + c' C — to' 

 i"=a"K + b'<B + d'C-m" 

 &c. 

 and A, B, C will be found from the equations of the minima, viz. 

 as f a'e + a"e" + &c. = 

 bi + b' i + b"e' + &c. = 

 c H + c e' + c"e" + &c. = 0. 



Again, let x ■= A + u 



y = 'Q ■\- V 

 z = C + W 

 and we may regard ii, v, w as the respective errors of A, B, C. 

 Substitute now the expressions of x, y, z, in the values of «•, 

 ^, e", &c., and we shall have 



e = s +au + bv + c'w 

 e' = e' + a' u + b'v + c'w 

 e'= e'' + a"u + b"v+ c" la 

 &c. 

 Further, put ><. = {au + b v+ c wy 

 + {a'u + b'v+ c wf 

 + {a"u-\-b"v-\-c"'wf 

 -i- &c. 

 then square the values of e, e^, e", &c. ; add the results into 

 one sum ; and leave out the terms which, on account of the 

 equations of the minima, are equal to zero ; we shall get 



S . ^^ = S . e' -1- A. 

 Now the probability of the function S .e* is equal to that of 

 the simultaneous existence of the errors whose squares are 



added 



