464 M. Cauchy on a New Formula for 
We should then take as the approximate value of y the 
arithmetical mean between the observed values; and the greatest 
error to be apprehended would be less for this than for any 
other approximate value. This property of arithmetical means, 
together with the facility with which they are calculated, com- 
pletely justifies the preference usually given to them in the 
valuation of those arbitrary constants which can be determined 
directly by observation. 
Let Ay be now what is wanting to complete the approxi- 
mate value of y furnished by the equation (12.), so that we 
have 
(13.), y=a8y,t+ Ay. 
Let us also put 
(14.) v=aSu,+ Av, w= «Su; + Aw, &e. ... 
we shall derive from the formula (3.) 
(15.) Sy, = aSu;+bS8v, + ¢Sw,, Xe... 
then from this last multiplied by « and subtracted from the 
equation (1.), we obtain 
(16.) Ay=bAv+cAw + &e.... 
Moreover, let us represent by «;, Ay;, Av;, Aw; what the 
values of a, Ay, Av, Aw, deduced from the equations (11.), 
(13.), and (14.), become when for 2 we substitute x;, 7 being 
one of the integers 1. 2...”. Ifthe values of Ay,, A yo --AYis 
are very small, and capable of being compared with the inevi- 
table errors of the observations, it will be useless to proceed 
to a second approximation, and we may rest satisfied with the 
approximate value of y afforded by the equation (12.). If the 
contrary takes place, it will be sufficient, in order to obtain a 
new approximation, if we do with the formula (16.) as in the 
first approximation we have done with the formula (1.). This 
being supposed, let us represent by 
S! A U; 
the sum of the numerical values of Av;, and by 
SAy;, S Aw; ..... Ke. 
the polynomials into which the sum S’ A a, is changed, when 
for each value of Av; we substitute the corresponding value 
of Ay; or of Aw; ...... « 
In fine, let 
Av 
ha a S'A 9; 
If we can without a sensible error disregard in the series 
the coefficient (c) of the third term and those of the following 
terms, we must take as the approximate value of Ay, 
