EXPRESSED IN TERMS OF THE ROOTS OP (pX, fx. 
451 
are equal. Hence universally ^=0, as was to be shown. To find r we must avail our- 
selves of the symmorphic, or as we may better say (it being- at the opposite extremity 
of the scale of forms, the antimorphic), value of ^ represented by \ i, taking care to 
preserve ^ strictly identical under both forms of representation, in point of sign as 
well as quantity. That is to say, we must make 
\i={- f" '^2 (j: - J {x - ...{x- 
Vhi 
h^ 
^?t+2' 
is- 
1 ‘Cj 
' S 
1 
^?i+l 
^?i+2‘ 
L 
1 
• 
* ■ 
where 
so that 
: ( - ) “2 (a: - ...{x- 
^?i+i 
^?i+2' 
1 
g 
gri 
hi 
h^ 
h 
• * 
'^1i + 2 • 
1 
• 
^ ^co ^ 'yni—i+mn—mi ^ 'ynn—i . 
and consequently the same reasoning as was applied to t to prove ^=G, will serve to 
show that — r=r, where 
hi ^2’ • - hm 
or 
r={-Yl{x- 71^) {x - ... (a: - 
h 1 h ^ ... hjii 
’Jfl • • • % 
where 
)—mn— 1 — mv—mn — 1 — m{n — I — 1) 
=mi—m — 1. 
Art. (30.). J have not succeeded in throwing t and r under any other than the 
single forms for each above given, and it is remarkable that whilst apparently t and 
r admit only of this single representation, ^ admits of the variety of forms included 
under the general symbol for a given value of i ; and it ought to be remarked 
that these forms (although the most perfectly symmetrical and exactly balanced 
representations) [and for that reason possibly the most commodious for the ascer- 
tainment of the allotrious factor belonging to them respectively] by no means exhaust 
the almost infinite variety of modes by which the simplified residues, i. e. the hekisto- 
barytic, or if we like so to call them, the prime conjunctives, admit of being represented 
