462 M. Cauchy on a New Formula for 
equations (5.) would, whether considered separately or in 
combination with one another, be all precisely equal, if the 
particular values of y which we suppose to be furnished by 
observation were rigorously exact. But they are not so; for 
actual observation is inevitably liable to errors confined within 
certain limits, and this consideration renders it advisable so to 
combine the equations among themselves that, in the most 
unfavourable cases, the effect produced on the value of the 
coefficient a by the errors committed in respect to the values 
of ¥;5 Yos «+» Yay May be the least possible. Now the different 
combinations that can be made of the equations (5.) in order 
to derive from them a new equation of the first degree in re- 
ference to a, will all furnish values of a comprised in the ge- 
neral formula 
(6.) pal Ie 9, + Ka Yo + acvccvere EnYn 
Ke Uy + keg Ug + corcovece kyUly 
which we obtain by adding together the equation (5.), mem- 
ber by member, and multiplying them respectively by the con- 
stant factors /,, k,...k,. It is still further to be observed, 
that, as the value of a determined by the equation (6.) does not 
vary, while we cause the factors /,, /, ... &, to vary simulta- 
neously in the same ratio, it is clear that the greatest among 
these factors (the sign not being taken into account) may al- 
ways be considered as reduced to unity. 
Finally, let it be observed that if we represent by 
£19 Eq coveee Eng 
the errors committed in the observations and the values of 
Y\ > Yo +++ Yn Yespectively, the formula (6.) will furnish an ap- 
proximate value of a, the difference between which and the 
true value will be 
Teves tis es oe. boc epee 
(7-) ag ha 
It is now necessary to choose /,, /,, .-- #, such that, in the 
most unfavourable cases, the numerical value of the expression 
(7.) may be the least possible. 
Let us represent by 
Su; 
the sum of the several numerical values of w;, that is to say, 
what the polynomial + w, + w+ ceseseeee + %q becomes 
when we so dispose of each sign in it that each term will be 
positive. 
Let us represent by Se, not the sum of the numerical 
values Of ¢,, &95 €3 «+ én) but what the sum Sw, becomes when 
