226 
Transactions of the Royal Society of South Africa, 
Treating in the same way with nainors of D3 we get 
2 
4 
6 
8 
0 
2 
4 
6 
m 
3 
5,1 
i 
1 
3 
5 
71 
^3 
d; d;'| 
Therefore, when D3 = 0, D^, and D5 have opposite signs. 
The proof will be seen to be general (for even n). 
The result may be thus stated : — 
Build up a determinant D„_i by means of concentric determinants 
Di = Ai; . Aj, A3 etc. Then we have proved that of every 
A„ A3, As 
three consecutive determinants the extremes have opposite signs when the 
middle one is 0. 
0, 2, 4 
Again when = A^ = 0, D3 
. 3 
3 5 
-0.3-. 
Now 0 is 7- f''(a) =ao, which we take to be +• 
/7^'' ^ ^ 
We will prove in subsequent articles that D^, D3, D^, ... D^, are of 
dimensions 1, 6, 15, 'u(u + l)/2 and that the coefficient of the highest 
power of a is always +. Hence the following scheme of signs and roots 
(0 on a line indicating a group of an odd number of roots) : 
a = +00 
— { 
f 
f 
A 
+ — - 
r 
2 groups of roots. 
- + 
■ + 
- + 
- + 
Degree. 
1 
6 
15 
3.7/2 
7z(w-l)/2 
