44 
Proceedings of the Royal Society of Edinburgh. [Sess. 
functions of the form uA-ffB whose degree-number is less than k must 
vanish identically, and that therefore in particular the last k “ simplified 
remainders” of A and B must so vanish. In the next place, proof is 
adduced that the vanishing of these remainders is equivalent to the vanish- 
ing of their first coefficients : and finally, there is reached the following 
variant to the above proposition regarding the highest-common-divisor : 
In order that the equations A = 0, B = 0 may have k common roots, it is 
necessary and sufficient that D n , D n _i, . . . , D n _ k+1 vanish : and, this being 
the case, the equation of the said common roots is p n _ k = 0. The fact that 
the vanishing of the first coefficients of the “ simplified remainders ” implies 
in each case the vanishing of all the coefficients following the first is merely 
commented on in passing. Attention, however, is more fully drawn to the 
important fact that the existence of the condensation-theorem makes it 
possible to put every proposition, which, like the foregoing, involves 
bigradients, into an alternative form. Thus, the condition that the 
equations 
x 3 + aqje 2 + a 2 x + « 3 = 0 t 
x? + b-jX + b 2 = 0 j 
may have two roots in common is, according to the said proposition, the 
vanishing of 
1 oq a. 2 a z 
1 oq a 2 ci s 
. . 1 \ b 2 
. 1 \ b 2 . 
1 \ b 2 . . 
and this by the condensation-theorem is the same as the vanishing of 
a 2 
1 
b 2 + cl-Jj 1 — a 2 
a l b 2 - a B 
oq& 2 — <^3 
Cltpbc) ? 
1 J h 
Zq b 2 -f* CL-fj-y ei > 2 | 
Bezout’s “ abridged method ” and Sylvester’s “ dialytic ” method, which 
resemble each other in involving elimination of successive powers of a 
common root, are only introduced by Trudi for purposes of corroboration. 
In connection with the former method there is noted Sylvester’s theorem * 
that the derived equations provide also an alternative way of obtaining the 
Sturmian “ simplified remainders,” the first remainder being the non-zero 
member of the first equation, the second remainder being the result of 
eliminating the highest power of x from the first two equations, the third 
remainder the result of eliminating the two highest powers of x from the 
* See Art. 5 of “ On a theory of the syzygetic relations . . .” 
