39 
1913-14.] The Theory of Bigradients from 1859 to 1880. 
The use to which this second theorem is put (pp. 132-137) is in 
connection with the division-process for finding the highest-common- 
divisor of two integral functions, and, in particular, with the modification 
of the said process employed by Sturm in obtaining his so-called 
“remainders.” From the general theorem* connecting dividend, divisor, 
quotient, and remainder we know that the coefficients of the first remainder 
in such a process are proportional to the successive determinants of a 
bigradient array composed of the coefficients of the dividend and divisor. 
We thus also know that this remainder having been made the divisor and 
the previous divisor the dividend, the new remainder must be expressible 
in like fashion. In the second bigradient array thus arising, however, one 
of the two sets of elements is complicated, being in fact the successive 
determinants of the previous array : and what Trudi’s “ dilatation ” theorem 
enables us to do is to supplant it by another array whose elements are 
simply the coefficients of the original functions. In this way the theorem 
finally reached is : The coefficients of the r th remainder R r arising in the 
course of the performance of Sturms division-process on 
a^x m + . . . + a m , b 0 x 11 + ... + b n 
are equal to the successive determinants of the array 
(a 0 , . . 
■ , a,n)r 
(b 0 . ■ . 
.,K) \ 
divided by the product of the squares of the first coefficients of all 
the preceding remainders and by b 0 m-n+1 and by the sign-factor 
( _ xive r th remainder, when divested of the threefold divisor 
here specified, a r say, Trudi follows Sylvester in calling the residuo 
semplificatofr and denotes by p r , so that R r = p r /a r . For example, when 
the originating functions A and B are 
ax i + bx B + ex 1 + dx + e, 
px* + qx 2 + rx + s, 
* See under Recurrents. 
t A most natural and helpful notation for such a remainder would he 
II Oo, . . . , a. m )r II ( x n ~ r ,. 1). 
II (&o, bn) II 
Thus, in the case here used for purposes of illustration, the remainders would be written 
a b 
c d e j 
(x“, x , 1), 
|| {abed 
e h II 
. p 
q r s \ 
l| (p q r 
s) 3 |l 
P <1 
r s . 1 
1 
