A Proof by Elementary Methods. 
219 
These seven take values . , . , . , . , D^, D'., qD^, where has the 
meaning ah^eady given to it in § 2. 
Thus, noticing that the original last row was multiplied by (04)g, which 
is seen to be O.Dgg, where D3 is the concentric determinant of order 3, 
we have 
0, 1, 2, 3, 4, 5, 68 
., ., ., D5, D;, 2D3 
Therefore, reading off from the last row, 
^?D7.D3 = g.D5.D5-D;.D; + D5.D;' (say). 
Therefore, when 1)5 = 0, qD^ . D3 is — ; that is, D^, D3 have opposite or 
like signs as ^ is +. 
Now treat D3 in the same way, using minors of D3 ; we get 
1, 
2, 
3, 
48 
0, 
1, 
28, 
37 
0, 
1, 46 
08, 
17, 
26 
where 1)3 = 
08, 35 
06, 
15 
•85! 
•74, 24 
I>3^ 
gDs.O.D.E 
gD3D, = ((?D3 + D;')D3-D;-' + D3.D; say. 
Hence, when 1)3 = 0, D3, Dj have opposite or like signs as g is +• 
Now 
and when 08 is zero 
D, = 08 = a< 
0: 
1, 28 
. , 17 
^, 06 
0-17^ 
Therefore when 08 = 0, D3 has the opposite sign to qr^ 
§ 5. This method, though for convenience the details have been given 
for 7z = 8, will be found to be quite general for 71 even. 
* A similar treatment reduces dialytic and circulant determinants to two or three 
terms of like form to those in the text. 
