WHICH SATISFY GIVEN CONDITIONS. 
95 
P', =Q, line-pair-point. 
Q', =P, line-pair-point. 
J', ut supra. 
R', inflexion of curve terminated by the inflexion tangent and by tangent to a curve : 
which being so, we have 
(3)(1), =(3) m (l) m , 
P =«)»„ 
2 
P' = Wj, 
Q= nn x . 
2 
Qj=mm x , 
J = /m,, 
5 
J' = 
R = Jtm,. 
4 
D'= m. 
(3, 1), =(3, l) m . 
P=25, 
2 
O 
II 
to 
Q=2 r, 
2 
P'=2i, 
J = i(m— 3), 
5 
co' 
1 
s* 
II 
1-9 
CO 
1 
J, 
II 
PS 
4 
R' = i{n — 3). 
46. And lastly, we have the point-pairs N, O ( line-pair-points ) and the line-pairs 
N', O' ( line-pair-points ), ut supra , , and 
( 4 ), ==( 4 ) m . 
N=i, 
0=z. 
N '=*, 
0 '=/. 
47. Where in all cases the central column of figures gives the numerical factors 
which multiply the corresponding capitals, thus we have 
for (1)(1)(1)(1) 
x=2i> —p=A + 2B + 4C, 
vr=2p—v =A'-}-2B'+4C' ; 
for (1, 1)(1)(1), 
\=2» -p= A +2B +4C +3D, 
■zs—2[h—v =A'+2B'+4C'+3D', 
and so on. 
48. The elements (m, n, S, r, /) of a curve satisfy Pluceer’s six equations, and 
Zetjthen uses these equations, in a somewhat unsystematic way, to simplify the form 
of his results. 
It is convenient in his formulae to write 3m + /, =3n + %, =a, and to express every 
thing in terms of ( m , n. a), viz. we have for this purpose 
2S = m 2 — m + 8n — 3a, 
2r=n 2 —Sm—n — Sa. 
