218 
Transactions of the Boyal Society of South Africa. 
(ii) The highest power of - is obtained from the 8's in the other 
8*7 
diagonal: its index is (7 + 6+... + !)= and its sign is 
( — — = (_)7'4^ and generally, if the degree of the original 
Ti(n— i) 
function is the sign is ( — — = ( — ) ' . 
We shall assume n to be even and and a^ to be + : for, if n is odd, 
or (Xo, a„ have opposite signs, a^x'' + o.^x''~^ + ... + a„ = 0 is known, on account 
of its continuity, to become 0 for some real value (a) or values of x : thus 
there are factors {x — a) : removing these, we have in every case only to 
consider the divisibility by (x^ —px + of a function of even degree, with 
a„ and 
Thus considering the determinant equation to be one in q' ^for it 
begins and ends, a^~'( — q') ' ... +ao'"~"=--0, so that the equation certainly 
has a real root unless 7i is " even-even." 
§ 4. Continuing the work for n = 8, we notice that is a determinant 
concentric with D^. 
Let (48), (37), (26), (15), (04) in brackets denote the minors of the 
corresponding elements of the last column of D^, and treat thus : — 
Multiply the 1st row by (48), and the 3rd by (48)g ; the 2nd and 4th 
by (37), {37)q ; 3rd and 5th by (26), {26)q ; 4th and 6th by (15), (15)g ; the 
5th and 7th by (04), (04)g ; and add columns for a new last row. 
The scheme is represented below, the multipliers being written against 
the rows, and being screened. 
1, 
2, 
3, 
4, 
5, 
0, 
1, 
2, 
3, 
48, 
0, 
1, 
28, 
37, 
08, 
17, 
26, 
•85! 
•74, 
06, 
15, 
•86, 
•75, 
•64, 
•53, 
04, 
.(48) 
.(37) 
.(26) + (48)2 
.(15) + (37)2 
.(04) + (26)2 
(15)2 
(04)2. 
The elements of the new last row are determinants consisting of the 
jQrst four columns of with varying fifth columns as shown below. 
0, 1, 
2, 
3 
0 
1 
2 + Og 
3 + 1^? 
48 + 2(/ 
57 + 3g 
48^1 
., 0, 
1, 
28 
0 
1 
28 + Oq 
37 + lg 
46 + 28^ 
S7q\ 
08, 
17 
08 
17 
26 + 08g 
35 + 17g 
26q' 
.! -83! 
06 
•85 
06 + '85^ 
15^'7,q 
24 + 06^ 
Ibq 
86, -7,, 
•64, 
•86 
•7. 
•64 + -86^ 
•53+ -75^ 
04 + -64^ 
13 + ^53^ 
04g 
