224 Transactions of the Boyal Society of South Africa. 
To eliminate (i^ from 
Ao/3" + A,/3'^-^+... + A, = 0 
A,/3"- + A3/3'^-4+...+A_ = 0 
and 
we multiply both equations by i^^, the upper one 
1 ) times, 
and the lower \^ — 1 j times, so obtaining (?i — 1) equations in (3^, /3^, . . . , fi"- 
from which we get a determinant eliminating /3. 
Thus for n-.8 
Ao/3- + A,/3- + A,/38 + A6/36 + Asi3^ = 0 
Ao/3 - + A,/38 + A,/36 + A6/34 + A8/3^ - 0 
Ao/38 + A,/36 + A,/34 + A6/3^ + Ag = 0 
A,/3^ + A3/34 + As/B^ + A7 =^ 0 
A,/3s + A3/36 + A5^4 + A,/3- = 0 
A,/3-+A3/38 + A5/36 + A,/34 = 0 
A,(3'^ + A3/3'° + A5/38 + A7/36 = 0 
and 
A., 
A„ A3, 
A„ 
A3, 
Ao, 
A„ 
A3, 
A., 
A„ 
A„ 
Agj 
A„ 
A3, 
A=, 
A„ 
A8, ., 
Ae, Ag, 
A„ 
A3, 
As, 
A_, 
Ae, Ag 
A55 A7, 
A,, . , 
= 0 = D,. 
(a dot meaning zero.) 
Also, omitting the first and last of the equations, we have 5 equations 
for the 5 "unknowns," /3^, /34, j3'°. 
Thus 
Ao> 
A^, 
A4, 
Ae, 
A„, 
A., 
K 
Ae, 
As 
Aq, 
A2, 
A„ As 
Aq, 
A„ 
A„ 
A6 
A„ 
A3, A, 
A., 
A3, 
A5 
A., 
A3, 
A3, . 
A., 
A3, 
A5, 
A, 
Ax, 
A3, 
A5, 
A„ . 
Ax, 
A3, 
A3, 
A„ 
D5' 
say. 
We proceed to prove that 1)^ = 0 has (at least) 4 real roots. We shall 
use the suf&xes of Aq, A^, etc., only. 
