552 J. D. HAMILTON DICKSON ON RELATIONS BETWEEN FUNCTIONS 
b , ba + 5c , 15bx® + 25ca +10d , ba’ + Sex’ +10dx* + 10eu’ + 5fa?+gx| (41). 
a, 6ax+5b , 15ax’?+ 25ba+10c , ax’ + 5bx? + 10cx* + 10dx’ + Sex’ + fa 
= by, dba + 5e , bu’ + 5ca* +10dax’?+10ex? + 5fx +9 
Pan as dax+ 5b, ax? + 5ba*+10cx’ + 10da’ + 5en +f 
To reduce this, subtract the 3d line multiplied by x from the 1st; and the 
4th line multiplied by x from the 2d; then, add the 2d line multiplied by x 
to the 1st; and the 4th line multiplied by x to the 3d; and finally, intro- 
ducing the factors independent of «— 
Gg, =. 1 lb) (e408), Ina ees 
LI (a), 5) , 10(0) , 
- » (4%), 5(¢), (9) 
(z), 5(6), (/) 
which agrees with the result obtained before. 
I shall write this last result in the form 
S3= mame 
a Ad Sly 969 Nd oe alain 
to 
the A’ always being function of x, and having all its last column zeros except 
the two lowest constituents, which are respectively (9) , (f). 
Let also those functions—called after their discoverer SYLVESTER, Sylvester's 
functions—be shortly expressed thus, e.g., 
{(a—B)’(a—yv)’(B—y)’(a—8) (c—e) (x—C)} by the symbol 2, . (44), 
the suffix indicating the number of roots which enter into the squared product 
of differences under the 2. Then the results arrived at may be collected in the 
form, 
> =6 4 S,=-42 
Ai Ai 
23=6 — Si= AP 
eae (45) oe (46) 
= =e: Bi As 
PF oe 6° ae S, aa a Na 
_gp At Ai As 
Bye Binge a fs) 
