53 
1913-14.] The Theory of Bigradients from 1859 to 1880. 
The bigradient form of eliminant is here used in the establishing of the 
proposition that if the coefficients a r , b r , be functions of the r th degree in one 
and the same variable y, the eliminant is of the (mnf degree in the same 
variable. Janni’s proof, though not quite so good as it might have been, 
is the more interesting. The eliminant being 
a 0 a i a 2 a 3 
• (Xq a x ^3 
\ b \ ^2 5 
he, in effect, multiplies the columns in reverse order by y°, y l , y 2 , y s , y 4 
respectively, and then divides the rows in order by y 4 , y z , y 2 , y 1 , y a 
respectively, thus obtaining 
V/ 
«i2/ _1 a 2 y~ 2 a sV~ 3 
«o a \V~ X 
\ y h 
hy 2 b i y b 2 
\y 2 \y 
y 3 
In this equivalent form the elements of the first two rows are all now of 
the degree 0 in y , and those of the last three rows are all of the degree 2, 
whence comes at once the desired result. 
It should be noted that the procedure shows each term to be of the 
( mnff degree in y ; in other words, that the eliminant is homogeneous. 
Also, dispensing in the end with y, we may deduce the isobarism of the 
eliminant, its weight being mn. 
Zeuthen, H. G. (1874): Madsen, V. H. O. (1875). 
[En Bemaerking om Beviserne for Hovedsoetningen om Elimination 
mellem to algebraiske Ligninger. Tidsskrift for Math. (3), iv. 
pp. 165-171.] 
[En Bemserking om Sylvesters dialytiske Eliminationsmethode. 
Tidsskrift for Math. (3), v. pp. 144-145.] 
Zeuthen repeats Salmon’s mode of 1859 {Hist., ii. pp. 373-374) of using 
Euler’s treatment of two integral equations in x which have more than one 
common root : he is, however, more detailed, and takes the number of roots 
to be p. 
