220 Transactions of the Boyal Society of South Africa. 
Thus we have the result : — 
Build up a determinant of order 71 — 1, with the successive concentric 
determinants. 
2' 
a^- 
<1' 
and so on. 
It will be found that of each set of three successive determinants, the 
extremes have for + values of q opposite signs when the middle one is 0. 
Also Dj = ao — a„^" = 0 {n being even) has a + root; and D3, have 
opposite signs when Di = 0. The beginning and end of D3, D^, ... are 
'D^ = al...-\-alq'- ', 'D^ = al...—alq' ^ ', 'D^ = al,..-\-alq' \ 
Hence the following scheme of + roots and signs (0 on a line indicating 
a group of an odd numher of roots). 
^' = 0 
2'= +C0 
+ - 
Degree. 
n 
2 
T 
(?^ — 1) n 
Thus D„_i = 0 has at least ^ (^^ — 1 + 1), i-e. ^ real + roots. 
. • . aoX"" + ^lic""' + . . . + = 0 has factors of the form (x^ —px + q). 
Q.E.D. 
Notes. 
A. If (a + f/3) is a pair of imaginary roots, a2+/3^ = one of the ^ real 
values of q. Thus the above scheme giving regions for these values of q 
gives some indication of the values of the moduli of imaginary roots. 
