252 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[406 
the equations of the tangent planes are 
p (3cE 4- 3e(7 — 4dD) + a (eA — 8dB) + t (dA — \2cB) = 0, 
p ( „ ) + <r' ( „ ) + t ( „ ) = 0, 
in all which equations we have 3ce — 2d 2 = 0; and if to satisfy this equation we write 
c : d : e = 2 : 3/3 : 3/3 2 , then the equations of the tangent planes become 
/3 3 (A/3 -8B) + 8 (3(7/3- - 4D/3 + 2E) = 0, 
p (3GB 2 - 4D/3 + 2E) + (o-/8 + r) (A/3 - 8B) = 0, 
p ( » ) + (<?'/3-t)( „ ) = 0, 
or the three tangent planes intersect in the line A/3— 8B = 0, 3C/3" — 4>D/3 + 2E = 0, 
which completes the proof. 
Reverting to the sextic locus, 
(ae + 4bd — 3c 2 ) 2 — 27 (ace + 2bcd — ad 2 — b 2 e — c 3 ) 2 = 0, 
considered as a locus in 4-dimensional space depending on the five coordinates 
(a, b, c, d, e), this has upon it the twofold locus 
ae — 4 bd + 3c 2 = 0, ace + 2 bed — ad 2 — b-e — c s = 0, 
say the cuspidal locus, of the order 6, and the twofold locus 
6(«c — 6 2 ), 3 (ad — be), ae+2bd — 3c 2 , 3 (be — cd), 6(ce — d 2 ) | =0, 
a , b , c , d , e 
say the nodal locus, of the order 4: there is also a threefold locus, 
a, b, c, d | = 0, 
b, c,d,e 
say the supercuspidal locus, of the order 4. We thence at once infer 
(IkI, 2 :) = 6, agreeing with (1/cl, 2 :) = a — 4, 
(bcl, 1, 1 :) = 4, „ „ (LtT, 1, 1:) = 2m 2 + 2mn + - 8m — + 13 — |a, 
(la, 3 :) = 4, „ „ (1*1, 3 :) = -4m-3w-5 + 3a; 
but I have not investigated the application to the symbols with • / or //. 
If the conic, instead of simply passing through the cusp, touches the cuspidal 
tangent, then in the equation (a, b, 0, f, g, K§oc, y, zf = 0 of the conic we have f— 0, 
or, what is the same thing, in the equation (e, 6c, 0, \a, 2b, 2d^cc, y, z) 2 = 0 of the 
conic we have a = 0. The equation in 0 is thus reduced to 4b6 3 + 6c0 2 + 4dd + e = 0. 
For the independent discussion of this case it is convenient to alter the coefficients 
so that the equation in 0 may be in the standard form (a, b, c, d\0, l) 3 = 0, viz. we