414]
ON POLYZOMAL CURVES.
493
and the four forms of the equation are found to be
( . , Vr(S — 7), Vo-(/3 — S), Vp (y —/3)) (Va'i/, Vb'F, Vc'TF, Vd / 2 7 ) = 0,
Vt (7 — S ), . , Vp (8 — a ), Vo- (a — y)
Vo- (¿>-ß), Vp(a-S), . , Vt (/3 — a)
Vp (£- 7), Vo-(7 — a), Vr(a-/8),
viz., these are the equivalent forms of the original equation assumed to be
(/3 — 7) Vpa' ¿7+ (7 — a) Vo-b' F + (a — /3) Vrc' IT = 0.
50. I remark that the theorem of the variable zomal may be obtained as a
transformation theorem—viz., comparing the equation \/lU+ VmF+ V?iTF= 0 with the
equation V&r+ Vmy + Vw^ = 0; this last belongs to a conic touched by the three lines
x = 0, y = 0, z — 0; the equation of the same conic must, it is clear, be expressible in
a similar form by means of any other three tangents thereof, but the equation of any
tangent of the conic is a# 4-by 4 02 = 0, where a, b, c are any quantities satisfying the
condition - 4- — 4- - = 0 ; whence, writing a# 4- by 4- cz + dw = 0, we may introduce w = 0
u c
along with any two of the original zomals x = Ü, y = 0, z = 0, or, instead of them, any
three functions of the form w; and then the mere change of x, y, 2, w into U, V, W, T
gives the theorem. But it is as easy to conduct the analysis with (U, V, W, T) as
with (x, y, z, w), and, so conducted, it is really the same analysis as that whereby the
theorem is established ante, No. 47.
51. It is worth while to exhibit the equation of the curve
y/lU + VmF + ViilF = 0,
in a form containing three new zomals. Observe that the equation - + ^ + - = 0 is
£1 D C
satisfied by a = l<f>x, b = c = n0cj>, if only 0 + (f> + % = 0 ; or say, if 0 = a' - a",
$ = a" — a, x = a ~ a "’ The equation
X V(a — a’) (a — a") IU + (a — a") (a' — a)mV+ (a" — a) (a" — a') n W
+ ^ V(6 -b')(b- b") lU + (b'~ b") (b' -b) mV + (6" — b)(b" —b')n W
+ v V(c - c'){g — c") IU + (c — c")(c' — c)mV + (c" — c) (c" — c')nW = 0
is consequently an equation involving three zomals of the proper form; and we can deter
mine X, /jl, v in suchwise as to identify this with the original equation ^IU + \ZmV+\/nW,
viz., writing successively U = 0, V= 0, W = 0, we find
(a' — a") X 4- (b' — b”) /x + (c' — c") v = 0,
{a” — a ) X + (6" — b ) fx + (c" — c ) v = 0,
(a — a') X 4- (b —b')fx + (c —c')v = 0,
