504] ON THE MECHANICAL DESCRIPTION OF CERTAIN SEXTIC CURVES. 143 
It would at first sight appear that the curve might become unicursal in a different 
manner; viz. it would be unicursal if 
16a 2 b 2 u 2 — (A + Bu 2 ) (G + Du 2 ) 
was a perfect square; but this is only the case when one of the four sides a, b, c, d 
is = 0. The condition in fact is 
AD + BG- 16a 2 b 2 = 2 \/ABCD; 
that is 
W'/x/jl' — vv pa — 16a 2 b 2 = 2 V — W' p,p! pa; 
where, putting for shortness M = d 2 — a 2 — b 2 — c 2 , we have 
XX' = M — 2bc + 2ca + 2 ab, 
¡jbfx = M + 2 be — 2ca + 2 ab, 
vv' = M+ 2 be + 2 ca — 2 ab, 
— p'a = ill — 2be — 2ca — 2ab ; 
and thence 
XX'/x/j,' = M 2 - W-c 2 - 4c 2 a 2 + 4a 2 6 2 + 4a&if + 8c 2 ab, 
— vv pa = M 2 — 4 b 2 c 2 — 4 c 2 a 2 + 4a 2 6 2 — 4 abM — 8 c 2 ab, 
and the equation thus becomes 
M 2 — 45 2 c 2 — 4c 2 a- 2 — 4a 2 6 2 = V (M 2 — 4 b 2 c 2 — 4 c 2 a 2 + 4 a 2 b 2 ) 2 — 16a 2 6 2 (M + 2c 2 ) 2 , 
viz. putting for a moment X = M 2 — 4c 2 (a 2 + b 2 ), this is 
(X - 4a 2 5 2 ) 2 = (X + 4a 2 b 2 ) 2 - 16a 2 5 2 (M + 2c 2 ) 2 , . 
that is 
16a 2 6 2 {X — (M + 2c 2 ) 2 } = 0; 
or, substituting for X its value, the equation is 
64<a 2 b 2 c 2 (M + a 2 + b 2 + c 2 ) = 0, 
that is a 2 b 2 c 2 d 2 = 0. 
We may have simultaneously 1°, a = d, b = c; 2°, b = d, a=c; 3°, c = d, a=b; the 
three cases are really equivalent, but the results present themselves in different forms. 
1°. Here A = 0, B = 4a (a -b), C= 0, H = 4a(a + 6); the relation between u, v 
contains the factor u, and throwing this out, and also the constant factor 4a, it is 
u [(a - b) + (a + b) v 2 ] + 2bv = 0, 
viz. u is given as a rational function of v.