408 Proceedings of the Royal Society of Edinburgh. [Sess. 
of Lagrange’s interpolation-formula, or the finding of a function u of the 
form N(o 5 )/M(os) which shall have the values u 1 ,u 2 , . . . , u n+m+1 when x 
has the values x x , x 2 , . . . . , x n+m+1 , it being understood that N and M are 
respectively of the n th and m th degrees in x. 
The given n -f m + 1 equations 
= N(aq) , u 2 M(a? 2 ) = JS"(^ 2 ) , 
are first used to eliminate the n -\- 1 coefficients of N( t x), and thereby obtain 
m equations for the determination of the ratios of the coefficients of M(a>). 
This is interestingly accomplished by using the multipliers x{\f\x ^) , x\ /f(x 2 ) 
. , where f(x)-(x — x 1 )(x — x 2 ) .... (x — x n+m+1 ) , then performing 
addition, and finally utilising a known theorem regarding “ partial 
fractions.” The result is that for any one value of p we have 
E=n+m+l i=n+m + 1 
Xj f\ x d ~ A x *) 9 
and that therefore when p has any one of the values 0,1,2, . . . , m — 1 
we have 
£=n+m+ 1 
2- 
i i x^M(x i ) _ q 
TW ~ 
By putting 
and 
V for X * Ul 4- X * U<1 -4- 
** for mm ' 
_j_ •^n+m+l^'n+m+l 
f { x n+m+ 1) 
a + a 1 x + a 2 x 2 + . . . + a m X m for M(a?) 
these last m equations become 
V 0 a + Wjoq + 
V 2 a 2 + . 
. . + V m (X m 
0 ' 
V Y a + V 2 a Y + 
V 3 a 2 + . 
• • T" = 
0 
^m-i a + v m a 1 + V. 
m+l«2+ • 
. . +«2iaH*m = 
0 
whence for M(ce) there is obtained the expression * 
* By making the observation that the v’s, are neatly expressible as determinants 
the whole matter may be put much more simply. Thus, taking the case where 
u = (/3q + &i%)/{a 0 + a 4 x + a 2 ^ 2 ), we see at a glance that 
1 x 4 xf apfa + faxi) 
1 X.2 X^ > (Bq + fiiXy) 
1 a? 3 a? 3 2 xfifo + faxa) 
1 x 4 x 4 . x^(p 0 + & 4 X 4 ) 
= 0 when p = 0 or 1 , 
[continued on next page. 
