1913-14.] The Theory of Bigradients from 1859 to 1880. 43 
has one of the Stnrmian properties which the p’s have been shown to 
possess. 
As regards the highest-common-divisor (pp. 142-144) his result is : 
In order that two functions may have a common divisor of the k* /l degree, 
it is necessary and sufficient that the first determinant of each of the last 
k of their bigradient arrays shall vanish: and, when this holds, the 
coefficients of the divisor in question are the successive determinants of 
the (n — k) <A array. For example, the functions being 
cLqX* + ape 1 + . . . + « 8 , bpx b + b Y x^ + ... + b b , 
their bigradient arrays are 
(« 0 . • • 
• > a s)\ 
1 (« 0 . • ' 
• • > “3)2 
1 ( a o > • • 
• . «s)j 
(«o . ■ • 
• ’ ^3)4 
(«<>>•• 
a 
00 
Cr< 
ft,,-. 
■ > h) 
, life,- 
• , h) 
(V- 
) 
(V-- 
• . h) 
and the proposition states that if the first determinant of each of the last 
three arrays vanishes, the functions have the common cubic factor 
(<*o> • 
X 
(60. • 
At a later stage (p. 151) there is given the supplementary proposition 
that the quotients resulting from dividing A and B by the said highest- 
common-divisor are, save for an unimportant factor in each case, the 
coefficients of B and A in Trudi’s form of the (n — k + iy /l “ simplified 
remainder ” — that is to say, are V n _* +1 and U n _ fe+1 as before defined. 
The closely related question concerning the common roots of two 
equations he deals with at length in a section devoted to elimination 
(pp. 161-178). Starting with the proposition that, u and v being integral 
functions of x, uA-f-tB must vanish for any common root of the equations 
A = 0, B = 0, he next points out that u and v may be so chosen as to make 
uA + tB of a low degree in x, even of the degree zero. In the latter 
extreme case uA + vB must contain the eliminant as a factor, and if in 
addition it be of the proper degree in the coefficients of A and B it is the 
eliminant pure and simple. Attention is then called to the fact that the 
division-process for finding the highest-common-divisor of A and B, or the 
Sturmian modification of this process, supplies a series of pairs of functions 
like u and v, and in particular that the last ££ simplified remainder ” D n , as 
satisfying all the requirements mentioned, is the eliminant. The condition 
for the existence of more than one common root is investigated in like 
manner. If the number of the roots in question be h, the degree-number 
of wA + tB cannot be less than h Founding on this, it is asserted that 
