PEOEESSOE CAYLEY ON EECIPEOCAL SUEFACES. 
225 
a{n— 2)= a — B + 
b{n- 2)= § +2/3+3y+3f, 
c(w— 2)=2ff+4/3+ y+3, 
a(w— 2)(w— 3)-=2(S— C)+3(ac— 3'<r— %)+2 (a 5— 2g — 
i(w— 2)(w— 3)=4& + (afl— 2g— j)+3(bc — 3/3— 27 — i), 
c(n—2)(n—3)=6h + (ac— 3c — %) + 2(fo — 3/3— 27 — i), 
q== b 2 -b-2Ic-3y-6t, 
r=c 2 —c—2h—3(3, 
a= n '(n'-l)-2b'-3c', 
x = 3 n'(n' —2)— 6b' — 8c', 
l=ln'(n'-2)(n' 2 -9)-(n' 2 -n'-6)(2b'+3c')+2b'{b'-l)+6b'c'+^c’(c'-l), 
a!(n'-2)= x'—B' + g +2a>, 
b'(n'~ 2)= g'+2j3'+3y+3^ 
c>'-2)=2</+4/3'+ y + 3', 
«'(+ - 2)(w f - 3) = 2(y - O') + 3 (a'c' - 3c' - yj) + 2(a'V - 2g' -/), 
J'(^-2Xw'-3)=4^ + {dV—24S j+3(b'c'-3($'-2y>-i'), 
d(n'-2)(n'-3)=6h' + (aV — 3c'— yl) + 2(b' c' —3(3' —2y' —i'), 
q '=V*-b'-2V-3y'-6t', 
r'—c' 2 — c ' — 2h! — 3(3', 
together with the equations for [3 and (3' . 
66. The symbols signify as follows; viz. 
n , order of the surface. 
a , order of the tangent cone drawn from any point to the surface. 
h , number of nodal edges of the cone. 
x , number of its cuspidal edges. 
b , order of nodal curve. 
Jc , number of its apparent double points. 
t , number of its triple points. 
c [ , its class. 
q , number of points where nodal curve is met by curve of contact of tangent cone. 
j , number of pinch-points. 
c , order of cuspidal curve. 
h , number of its apparent double points. 
r , its class. 
c , number of points where cuspidal curve is met by curve of contact of tangent cone. 
mdccclxix. 2 1 
