45 
1913-14.] The Theory of Bigradients from 1859 to 1880. 
first three equations, and so on. In other words, if the set of equations 
derived from A = 0, B = 0 by Bezout’s method be 
c ll x m ~ 1 + c 12 x m ~ 2 + ... +c lm = 0 ^ 
c 21 x m_1 + C 22 x m ~ 2 + . . . +C 2m = 0 |- 
then the second, third, . . . “ simplified remainders ” of A and B are 
11 C l2 Xm " "f C 13^ m 3 ’ 
• • + c m 
21 ^22*^ "" "t“ ^23*^ ' 
. . + 
i r r r x m ~ 3 4- 
C 11 c 12 '13*^ ^ - 0 
• + C im 
^21 ^22 ^23*^ + , . 
• ^2 m 
C 31 ^32 C S2p C "t • • 
• +C 3 m 
Proceeding from this, Trudi then says that if the non-zero members of the 
said derived equations he denoted by Y v Y 2 . . . , the “ simplified remainders ” 
can clearly be put in the form 
V. 
c n Y, 
C 11 
C 12 Yj 
r Y i 
°21 x 2 1 * 
C 21 
c Y 
C 22 X 2 
C 31 
r Y 
c 32 x 3 
and, as by definition 
Y x — cLqB ’ 
Y 2 = (u 0 x + aq)B - (b 0 x + 6j)A , 
Y 3 = ( a ^x 2 + a x x + a 2 )B - (b Q x 2 + b x x + & 2 ) A , 
it follows that the said remainders have still another form, namely, 
0 
1 
CO 
■ ^0 
a o 
P> - 
c n 
K |A, 
| c 21 a 0 a; + oq 
C 21 
i<r 
+ 
& 
0 
C 11 C 12 a o 
P> - 
C 11 
C 12 ^0 
c 2X c 22 *t* oq 
^21 
^-'22 b 0 x -)- b j 
Cgi ^'gq a^x 2 4 - (x-^Kj 4- cl 2 
C 31 
Cg 2 b^x 2 -{- byC + b 2 
— a result easily shown, by the use of the condensation-theorem, to be in 
agreement with a previous one in which the determinant coefficients of A 
and B are bigradients. He is also careful to note that although here, as 
usual, n is taken equal to m, no real restriction is thereby made, the case 
where m>n being viewable as a case in which the coefficients of x n+1 , x n+ 2 , 
. . . , x m in B are equal to 0. For example, if the given equations be 
ax 4 + bx 3 + ex 2 + dx + e = 0 i 
qx 2 + rx + s = 0 f 
