410 Proceedings of the Royal Society of Edinburgh. [Sess. 
so that if we put 
we have 
R p for , 
~f(x) ~ aK o + a i R i' f ' • • • + J 
and therefore, by substituting the already found values of a\a^:a 2 
N(a?) 
A*) 
= 
R 0 
*1 • ■ 
Rm 
v i • ■ 
■ • v m 
. . 
V m+ 1 
|m-l 
v m . . 
v 2m-l 
As, however, xR p + v p = 1 R, p+x , we can change the elements of the second row 
here into R x , R 2 , . . . , R m+1 , and then the elements of the third row into 
R 2 , R 3 , . . . , R m+2 , and so on, thus arriving at a determinant of the same 
special form as in the case of M(a;) . 
Combining the two results, Jacobi is thus led to the theorem that 
l 
X(x) 
Ro 
Bj • ■ 
• Rm 
w o 
• * 
• w; m _! 
/(*) ' 
M(jb) 
E i 
r 2 . . 
Rm+1 
W 1 
W 2 • • 
• W m 
e 2 
R 3 . . 
Rm+2 
U>m- 1 
W m • 
• • W 2rn-1 
Rm 
Rm+1 • • 
Rjm 
— a result not easily verifiable by giving x one of its n-\-m + 1 values.* 
* Continuing the case of the previous footnote we should prefer to begin with 
N(aj) 
7&j 
I rvt 0^y» 1/y* *2/ y» 3 I 
JLl ^2 «^3 •*'4 I 
1 
Xi 2 
N(x x )/(x flJj) 
1 
a? 2 
x 2 
]$(x 2 )/{x-x 2 ) 
1 
x s 
X* 
N(x 3 )l(x-x 3 ) 
1 
X 4 
xf 
■N(aj 4 )/(aj-aj 4 ) 
and then proceeding exactly as before we should arrive at 
N(s) 
f(x) 
I rp 0 ry» 1 rp 2/y* 3 I 
| * 
Po 
Pi 
P2 
Pi 
P2 
Ps 
P2 
P3 
P4 
where p p = | x x ° x^ x 2 xj>uj(x -x 4 )\. 
The function sought would then be 
Po 
Pi 
P2 
( x - x 1 )(x- x 2 )[x - x s )(x - x 4 ) 
Pi 
P2 
P3 
P2 
P3 
P 
\x 1 °x 2 i x./x 4 i \ . &>o 
"1 
"1 
a>2 1 
