solving the Problem of Interpolation. 463 
in it we substitute for each value of u; the corresponding 
value of <, If we reduce to +1 or to —1 each of the co- 
efficients /,, k,, ... k, by so choosing the signs that in the 
denominator of the fraction (7.) all the terms may be positive, 
this fraction will be reduced to 
Se. 
Su;’ 
and it will afford a numerical value, at most, equal to the ratio 
Sa if we represent by the sum of the numerical values of 
é or, in other words, that of S<; in the case which is least 
favourable. On the other hand, by assigning to h,, k, ... Irn 
unequal values the greatest of which (the signs not being taken 
into account) may be unity, we shall obtain for the denominator 
the fraction (7.) a quantity whose numerical value will evi- 
dently be lower than Swz;, while that of the numerator may 
ascend even to the limit 3: and this will actually happen if 
the errors ¢;, £5 +++ & be all of no amount, except that one 
which is multiplied by a factor equal (the sign being disre- 
garded) to unity. Hence it follows that the greatest error to 
be apprehended in respect to the value of a determined by 
means of the formula (6.) will be the least possible if we put 
generally eco 
choosing the signs in such a manner that, in the polynomial, 
Ke, ty, + ig tty + veveee + ky Up, all the terms may be positive. 
Then the formula (6.) will give 
Sy 
(9.) lier S u; ; 
(Sy; being what the sum Su; becomes when in it we substi- 
tute for each value of wu; the corresponding value of y,,) and 
the equation (4.) will become 
(10.) a 
u 
Su; Sys 
If, as an abbreviation, we put (11..) @ = ce » we shall 
have 
(12.) yre Sy. 
If we supposed generally « = 1, the equation (4.) reduced 
to y = 0 would indicate that the value of y is constant ; and 
as we should then have 
u 
2£>a 
Su; 
l F ] 
= >, the formula (12.) would give y = a Sy. 
