MaleVs Proof that every Equation has Boots. 
229 
Therefore 
D„ beco.es (-)^^ >^y^);(y:^)3;;(y - p^od. of m. 
which, since - 1, v - 3 ... are even, and 2v - l<n + v -1, is +. 
Thus the signs assumed when a= + ao are justified, and the proposition 
is fully proved. 
§ 5. Note. — It can also be proved directly from the foregoing that /3 is 
a pure imaginary when f{x) does not vanish for real values of x. 
If we write in the equivalent form 
8 6 
4 
2 
0 
• 
8 
6 
4 
2 
0 
8 
6 
4 
2 
0 
7 
5 
3 
1 
7 
5 
3 
1 
7 
5 
3 
1 
7 
5 
3 
1 
the concentric determinants will be different. 
The new Dj will be seen to be equal to 
2 4 6 8. 
0 2 4 6 8 
.13 5 7 
1 3 5 7. 
3 5 7.. 
Dj is now 7, which =/'(a) ; and when 7 (of odd degree) = 0, D3 = - 8-5^, 
i.e. D3 has the opposite sign to /(a). Therefore, if f{a) is always -\- the 
scheme of signs is the same as in the previous work in § 2 (end). 
It follows that when = 0, the new has the same sign as the old 
D5 ; but the simultaneous equations which gave /3^, /34, ... /3'° show that 
^^Xr^^ /3°^' is - ve ; therefore /B^ is - ve, and /3 is imaginary. 
