42 
Proceedings of the Royal Society of Edinburgh. [Sess. 
showing that p r _ 2 and p r have different signs for any value of x that makes 
Pr^ vanish. 
Proceeding from the above-noted Cayleyan mode of expressing the 
“simplified remainders,” Trudi puts forward (pp. 145-152) another mode, 
each remainder now appearing as a sum of a multiple of A and a 
multiple of B ; or, in Sylvester’s words,* as a syzygetic function of A and B. 
For example, the three remainders above given he considers in the form 
1 
A + 
a b 
. p 
B 
p 
<1 • 
p 
q 
X 
a 
b 
c 
d 
X 
A + 
a 
b 
c 
d 
a 
b 
c 
1 
a 
b 
c 
V 
<1 
P 
q 
1 
V 
<Z 
r 
V 
q 
r 
X 
p 
Q 
r 
s 
p 
q 
r 
8* 
X 2 
and generally he writes 
Pr I U r .A + V r .B , 
where it is readily seen that as regards x 
- 1 , 
U r is of the degree r 
V, 
U r A 
y,b 
and where, as we know, 
Pr 
m — n + r 
m + r - 1 
m + r 
h 
r— 1 f 
n — r . 
Observation also shows that the coefficients of the highest powers of x in 
U r , V r are H^D^, afi r _ x respectively. By substituting the new forms 
for p r _ 2 , p r _ v p r in Cayley’s relation 
By— l * Pr— 2 Ct T _ 20 t T Q r * Pr — 1 “t ' p r 0 , 
there is obtained 
(D^Lj • U r _ 2 — a r _ 2 a r Q r ■ U r _ 1 + D^_2 • U r )A 
+ ( • V r _ 2 — a r _ 2 a r Q r • V r _ x + D^_ 2 • V r )B = 0 ; 
and, as Trudi proves that this can only hold when the coefficients of A and 
B vanish, it follows that each of the two series of functions 
B"o 5 1^1 5 • • • 5 B"n+1 5 V 0 , Y"i , . . . , y n+ l 
* It is worth noting that it was in this connection that the word “ syzygetic ” was first 
used, the full title of the memoir of 1853 (which clearly had considerable influence on Trudi) 
being “ On a theory of the syzygetic relations of two rational integral functions, comprising 
an application to the theory of Sturm’s functions, and that of the greatest algebraical 
common measure.” 
