MaleVs Proof that every Equation has Boots. 227 
It remains to justify the signs of D^, D3, ... when a= ±00 . We shall 
then have proved that D„_, = 0 has (at least) ^ real roots in a, and that 
has one value for each root of a. . • . f{x) = 0 has ^ pairs of roots (a + ^). 
Q. E. D. 
§ 3. To prove that the highest power of a in has index v{v-\-l)l2 
and a + coefficient. Eejecting all but highest powers in Aq, A^, taking 
to be 1, and indicating binomial coefficients by n^, 
D, = i 1 
n^a^, W^a^, Wga^, Uga^ \ 
1 , n^a^, n^a"^, Uf^a^ 
. , n^a, n^a'i, n^a^ 
UjU, n^a^, n^a=, nyci^ 
7?,o3, n.a^, n^a7, n^a^ 
Dividing out powers of a in the columns 
= a'+^+-^ 1 , n^a^, n^a^, n(,a^, WgaS 
1 , n^a, n^a, n^^a 
. , Wj, n^, n^ 
Uj^a, n^a^, n^a^, n^a^ 
! n^a, n^a3, n-a"^, W^a^, 7lga'^ 
Multiplying the first two columns by a3, a, we obtain 
1 , 7^2, n^, ng \ 
. , 1 , 71^, n^, Uf, 
. , n^, n^, n^, n^ 
llj, 71^, 71^, 7lg 
The index of a is 5(5 + l)/2, and a similar process gives for the highest 
index in v{v-\-l)l^. 
4. To prove that the coefficient is +. 
Consider D„= w^, 7?^, n^, 
n,-,+„ n,_,^„ n,_,^,, 
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