370 Prof. Sylvester on the Historical Origin of the 
quadratic equation ; for using ¢, f, F to denote rational symme- 
trical forms of function, it follows that . 
a, b, be, 
F 3 I pine . ee, d is itself a rational symmetric function 
pdt , of a, b, ¢, d. 
tba, d, pb, c) 
Whence it follows that if a, b, c, d be the roots of a biquadratic 
equation, f(a, b, ded) can be found by the solution of a cubic: 
for instance, (a+b) x x (c+d) can be thus determimed, whence 
immediately the sum of any two of the roots comes out from a 
quadratic equation. 
“To the modulus 6 there are fifteen different synthemes 
capable of being constructed. At first sight it might be sup- 
posed that these could be classed in natural families of three or 
of five each, on which supposition the equation of the sixth degree 
could be depressed ; but on inquiry this hope will prove to be 
futile, not but what natural affinities do exist between the totals ; 
but in order to separate them into families, each will have to be 
taken twice over; or in other words, the ‘fifteen synthemes to 
modulus 6 being ‘reduplicated, s subdivide into six natural families — 
of five each.” 
The six families above referred to (in which it is to be under- 
stood that p.g and q.p are identical in effect) are the followmg :— 
a0 Ca err Ore. ee ye aid exp ee 
G6 OLE) a.) ese V6.7 tere Ge GOA Pe 
We OL ewe ce Qe C.0 C.F af ad, t exe 
ave On eer bl ea Ee) a.0° af Bae 
Gop Ole We ATO ce Ce 4.0 GON 
ae f.6 ‘e.d Gf) bie die &.8) c,d oeef 
aifiivceve. Gvd a8. f.d5 exe a.c b.e d.f 
Gb eds fre a.c f.e b.d ad b. frie ce 
a.ee.b fid ad» fre bie a.6@ bd -0.f 
aud ef Bie a.e) f.b ed af bie! dle 
And it will be observed that every two families have one, and 
only one, syntheme in common between them ; and precisely m 
the same way as in the note above quoted, it is especially shown 
that the one single natural family 
a.b ec.d 
a.c b.d 
aya be 
gives rise to a function of four letters with only one value, so 
the six functions analogously formed with these six families ob- 
viously give rise to six functions, which change into one another 
