186 
ON THE MECHANICAL DESCRIPTION 
[446 
we have the locus as the inverse of a conic. To exhibit it as the antipode of a conic, 
taking X, Y as current coordinates measured from the node as origin, the equation of 
the line through a point of the locus, at right angles to the radius vector from the 
node, is 
X (x - a) 4 Y (y - /3) - O - a) 2 - (y - ft) 2 = 0; 
or, substituting for (x — a), (y—ft) their values, this is 
X[(l — iqiq) (u 2 + to) + (1 — to) (tq + u 2 ) u] 
4 Y [(X + u 2 ) (u 2 + m) - (1 — to) (1 — u x uf) u] + 2 (to 4- 1) a {u — iq) (u — u 2 ) = 0 ; 
and the antipodal conic is thus the envelope of the line represented by this equation. 
Putting for shortness 
P = X (1 — u x uf) 4- Y (zq + u 2 ), Q = X (zq + u 2 ) - Y (1 — zqw 2 ), 
the equation is 
tf{P+2(m+ 1) a} 4- u {(1 - to) Q - 2 (to + 1) a (zq + u 2 )) + mP + 2 (to + 1) a zqiq = 0, 
and the equation of the conic therefore is 
4 {P + 2 (to + 1) a] [mP + 2 (m + 1) a zqzq} — {(1 — to) Q — 2 (m + 1) a (zq + zq)} 2 = 0, 
so that the conic touches each of the lines P + 2 (to 4-1) a = 0, toP + 2 (to 4-1) a zqzq = 0 
at its intersection with the line (1 - to) Q — 2 (to + 1) azq — 0. If these lines were con 
structed, one other condition would suffice for the construction of the conic. 
The before-mentioned equations 
give 
and thence 
a m +1 
to (zq 2 4- 1)(m 2 2 + 1) 
a to + 1 
to (iq 2 + 1) (m 2 2 + 1) 
a 2 (to +1) 2 
to 2 (tc 2 +1) (u-2 2 +1) 
(1 — zq u 2 ) (to -f zqzq), 
(zq + zq )(TO + zqzq), 
(to 4- zqzq) 2 ; 
a to 1 — zqzq 
ct 4- ft 2 (to + 1) a to 4- zq iq ’ 
ft _ TO zq 4 zq 
a? + ft 2 (to 4-1) a to + zqzq ’ 
which determine zq + zq and zqzq rationally in terms of a, ft. For the cuspidal curve, 
writing zq = u 2 — v, we have 
a to 1 — v 2 
a 2 + ft 2 (to + 1) a to 4- v 2 ’ 
ft to 2v 
a 2 + ft 2 (to 4-1) a to 4- v 2 ’