426 
Proceedings of Royal Society of Edinburgh. [sess. 
( _ f )w— (/+0+^+ • • • +0^ 
we have on leaving out the common factor ( - 1) M the identity 
o = 2(-i y+9+ h + : • 
+i 
(flglhl 
n ! 
. . I \){l f P?> h . . . n l ) ’ 
which like it’s companion may be illustrated by the case of n = 5, 
viz., 
0 = 1 -10 + 15 + 20-20-30 + 24 * 
Lastly, attention is directed to the fact that when n is a prime, and 
therefore not exactly divisible by any integer less than itself, the 
number 
n ! 
(/! g\h \ . . . Z !)(1 / 2 ? 3 A . . . n l ) 
must be exactly divisible by n , except in the case 
/= n , <7 = 0, h = 0 , . . . 1 = 0 , 
when it has the value 1, and in the case 
/= 0, g = 0, h = 0, . . . 1=1, 
when it has the value (n-1) ! It, therefore, follows from either 
of the two preceding identities, that the sum of these two values 
must be divisible by n , — which is Wilson’s theorem. 
The remaining two pages are occupied with the expansion of a 
determinant of special form, viz., that afterwards known by the 
name axisymmetric. 
JACOBI (1841). 
[De formatione et proprietatibus Determinantium. Crellds Journal , 
xxii. pp. 285-318.] 
The value which Jacobi attached to determinants as an instru- 
ment of research has already become well known to us : we have 
* In connection with this and in illustration of a previous remark regarding 
a mode of expressing the full expansion of a determinant, we have 
2 ± &00 a ll a 22 a 33 a 44 = ®00 a ll a 22®33 a 44 — 2 « 0 0 a ll a 22 a 34 a 43 
+ 2a 00 a 12 ®21 a 34 a 43 d" 2 2^(^23^34*42 
— 2<X 01 tt2o*23*34*42 — 2a 0 o*i2*23*34*41 
+ 2 a 01 a 
12 * 23 * 34*40 • 
(LV. 2 ) 
