Mr. Ivory on the Method of the Least Squares. 83 



will therefore be always positive, and it will become infinitely 

 great whenever any of the errors is equal to ± cs=. There 

 will therefore be a minimum relatively to every error, and an 

 rbsolute minimum for certain definite values of all the errors. 

 The equations of the several minima are respectively, 

 d^ ifl , 1± . Ill + &c. = 0, 



de^ ■ dx ^ de" dx ,q^ 



i± ifl + ll.i^ + &c. = 0, 



de* ; dy ^ def^ dy 



and all these equations together determine the particular 

 values of x,y, &c., in the case of the absolute minimum.. 



Let us now suppose that the most probable values of 4^ are 

 the several minima, and consequently that the absolute mini- 

 mum is the most advantageous, or the most probable, value 

 7^11: then, since the probability of 4^ - J"st the same as 

 that of the simultaneous existence of the en'ors which enter 

 into it, that probability will have the product (P) ^^ ^ ex- 

 pression. Wherefore, in the supposition we have made, it 

 Llows that the minima of one function will take place at the 

 same time with the maxima of the other; and hence we get 



these formulae, viz. 



1 d.<>( e-_)_ __ ^9 ±i_ 



"^(e^) • de» • de«' 



1 d.(p{e<'^) _ ^2 ^ <^^ 



&C 



which render the equations (B) and (C) identical, all the lat- 

 ter being first multiplied by the arbitrary quantity A . Now 

 kis manifest that these formulae cannot be satisfied unless 4^ 

 liave this form, viz. 



4, =/(.') +/(^"0+/(^'") + &c.; 



in which case all the formulae are contained in one, viz. 



_1 djif) __ j^^ dfje"') 



(f (e'O ' de'^' • d e» ' 



which determines the function ^. 



If the probabilities of the several erjors ., ^, ^, &c. be ex 

 pressed by different functions, viz. <^{^\fV% ^ (^ )' &c., it 

 will follow that 4* must have this form, viz. 



and then the several formulae will determine the functions 



'^' t! iti'onler to apply the foregoing reasoning to a system of 

 equations of c.idiK we must recollect that the mo^t^ad^^^ 



