Mr. Ivory on the Method of the Least Squares. 9 



we liave three sets of weights, viz. a, a, a", &c., and ^, b , b'\ 

 &c., and c, c', c'', &c. ; and the levers e, e, e", &c. must he in 

 equihbrio when each set separately is suspended from their 

 extremities. The equations necessary for tliis purpose are 

 these three, viz. 



ae + a' e' -\- a" e" + 8u;. = 



be + b'e' + 6"^" + &c. = 



ce + c' e' + c"e" + &c. = 0, 

 which are sufficient for finding the corrections x, y, z. The 

 same equations may be thus written, viz. 



de , de' , _ de" II .-, 



-j-e + -r— t-' + -T— e" + &C. =0 

 ax dx di 



-r- ^ + -J— e + -r-^ + &C. =0 

 dy dy dy 



de de , . de" „, „ 



-T- e + -— e' + —- e + &c. =0, 



d z dz 11 z 



and these determine the minima of the sum of the squares of 

 the errors relatively to the variables x, y, z respectively. The 

 same three equations determine the absolute minimum of the 

 function relatively to all the variables ; and therefore, in this 

 single condition the full solution of the problem is contained. 

 It is evident that the same mode of reasoning will apply to 

 the simultaneous correction of any number of elements by 

 means of a system of equations of condition. 



The procedure which we have demonstrated has very pro- 

 perly been called the method of the least squares, since the 

 absolute minimum of the sum of the squares of the errors 

 contains all the conditions necessary for finding the several 

 corrections. It was first published by M. Legendre in his 

 treatise on the Orbits of Comets. But some years before the 

 publication of that work, M. Gauss, of Gcittingen, had like- 

 wise found out the same process, which he had communicated 

 to his astronomical friends ; and he was in the habit of apply- 

 ing it in astronomical researches. 



We have here attempted to demonstrate the method of tlie 

 least squares, from the nature of the equations of condition, 

 and from the principle that, in the most advantageous solu- 

 tion, the errors of the equations must be contained within the 

 least limits. But it has been usual to introduce the doctrine 

 of probabilities in order to explaiji this theory. A \^\X\q '^i- 

 tention will show that all such investigations are founded on 

 arbitrary sii})positioiis. We cannot compute the probability 

 of any combination of the errors, without first assoming a 

 general function fur expressing the probability of an error 

 taken indefinitely. There are many proixjties which such 



Vol. fi.'J. No. .'$21. .7«;7. IR2.5. " B a fiuic- 



