solving the Problem of Interpolation. 467 
(1X.) Sa; => 15 SB; = UV; S' B; => hs Sy: = Vs Sy: =U; 
S’ %= its SEC), 
but also 
(X.) SAv,= 0; SAw,; =v; § A? w;= 0; S'A* vw; = v, &c., 
and 
(XL) SAy,=0; SA’*y; = 0; S A?y; = 0; S'Aty;= 0.., 
These latter formule are so many equations of condition 
which must be satisfied by the particular values of a, 6, y... 
as well as by those of the several orders of differences of 
W, V, W... y; and hence it follows that in the calculation of these 
particular values we cannot commit an error of a single figure 
without being apprised of it by the bare fact of the equa- 
tion of condition ceasing to be verified. 
The advantages of the new formule of interpolation are the 
following: 
Ist. They are applied to the development by series, what- 
ever be the law according to which the different terms are 
deduced from one another, and whatever be the values, equi- 
different or not, of the independent variable. 
2nd. The new formule are of very easy application, espe- 
cially when logarithms are employed in the calculation of the 
ratios @, 8, y ... and in that of the product of those ratios by 
the sums of the several values of the functions or their differ- 
ences. Then, in fact, all the operations are reduced to addi- 
tions and subtractions. 
3rd. By means of these formulz the successive approxima- 
tions are made with a constantly increasing facility, as the se- 
veral orders of differences continually decrease. 
4th. They allow us to introduce at once into the calcula- 
tion the numbers furnished by all the given observations, and 
thus to add to the exactness of the results by making a great 
number of experiments subservient to this object. 
5th. They possess this advantage also, that, on every new 
approximation, the values which they furnish for the coeffi- 
cients a, b, ¢ are precisely those in which the greatest error to 
be apprehended is the least possible. 
6th. Our formule: indicate of themselves the moment when 
the calculation ought to cease by then giving differences com- 
parable with the errors of observation. 
7th. The quantities which they determine satisfy equations 
of condition which do not allow the least fault of calculation 
to be committed without being almost instantly perceived. 
In the new mathematical exercises there will be found nu- 
merous applications of the formule of interpolation. I shall 
here quote but one of them. 
Let / be the length of a luminous undulation relative to one 
$AQ 
