ON POLYZOMAL CURVES. 
550 
[414 
In fact, these values obviously satisfy the second equation; and to see that they 
satisfy the first equation, we have only to write them under the form 
p : a : t = M — 4 (b + c) (a + d) : M — 4 (c + a) (b + d) : M — 4 (a + b) (c + d), 
where M = (a + b + c + d) 2 . The first set gives for the curve 
(b - c) VSl + (c - a) + (a - 6) Vi = 0, 
but this contains the line 2 = 0 not once only, but twice; it in fact is (y°- = 0), the 
axis taken twice; the only proper cubic with the foci A, B, C, D in lined is therefore 
(b — c) (a + d — b — c) V31 + (c — a) (b + d — c — a) Vi8 + (a — b)(c + d — a — b) Vi = 0, 
the equation of which is, of course, expressible in each of the other three forms. 
Article Nos. 188 to 192. Case of the General Circular Cubit. 
188. Returning to the general case of the circular cubic, the lines BG, AD meet 
in jR, and if we denote by a u b 1} c lt d 1} the distances from R of the four points 
respectively, so that b-fi x = afh = rad. 2 R, then observing that a, b, c, d are proportional 
to the triangles BCD, CD A, DAB, ABC, with signs such that a + b + c + d = 0, we find 
a : b : c : d = — d 1 (b 1 — c l ) : c 1 (a 1 — d 1 ) : —b 1 (a 1 — d 1 ) : a l (b 1 — c l )\ 
and this being so, the equations - + y + - = 0, fl + Vm + fn = 0, give two systems of 
cl O C 
values of fl : Vm : Vw, viz., these are 
and 
fl : Vm : fn — b x — Cj : c 1 — a 1 : a 1 — b 1 , 
— b\ C\ . Ci "I - cq . a j b\. 
(To verify this, observe that for the first set we have 
l_ + m + n _ (¿>i - Ci) 2 + (Ci - aff (a, - 6Q 2 
a b c -d 1 (b 1 — Ci) Ci («j — dd) — b x (a x — d x ) ’ 
h-ci, L 
— d x «j — d x 
h Ci + h ~ c j ( a i 2 
d x a i d x \ ^i ci 
_ b x - Ci b x — Ci fa x 
d x ct x d x \d x 
0; 
and the like as regards the second set.) 
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