134 
PROFESSOR CAYLEY ON THE CURVES 
the equations of the tangent planes are 
f (3cE+3cC-4dD)+c(cA-8dB)+r(dA-12cB)=^ 
g'( » )+*'( » )+^( » )=°> 
in all which equations we have See— 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( 3C/3 2 — 4D/3 + 2E) = 0, 
( 3CB 2 — 4D/3 + 2E) -|- (<r /3 + r)(A/3 — 8B) = 0, 
§'( » )+(o-'/3-rX „ )=0, 
or the three tangent planes intersect in the line A/3— 8B=0, 3Q3 2 — 4D/3 + 2E=0, 
which completes the proof. 
Reverting to the sextic locus, 
(ae + ibd — 3c 2 ) 2 — 2 7 {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 — ^bd -f- 3c 2 = 0, ace -\-2 bed — ad? — b 2 e — c 3 = 0, 
say the cuspidal locus, of the order 6, and the twofold locus 
6(ac— b 2 ), 3 {ad— be), ae-\-2bd—Sc 2 , 3 {be—cd), Q(ce—d 2 ) II =0, 
a , b , c , d , e 
say the nodal locus, of the order 4 : there is also a threefold locus, 
«, b , c, d 1=0, 
b, c, d, e | 
say the supercuspidal locus, of the order 4. We thence at once infer 
(1*1, 2 : )=6, agreeing with (1*1, 2 : )=a— 4, 
(1*1, 1, 1 : ) = 4, „ „ (1*1,1, 1 : )=2m 2 + 2wm+^ 2 — 8m— fw + 13— fa 
(1*1, 3 :)=4, „ „ (1*1, 3 :)= — 4 m — 3 n — 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 , lifac, y , :s) 2 =0 of the conic we have f= 0, or, what is 
the same thing, in the equation ( e , 6c, 0, fa, 2 b, 2 dyjc, y, z ) 2 = 0 of the conic we have 
a=0. The equation in 6 is thus reduced to ib8 i -\-§c0 2 -\-A.dQ-\-e=0. For the inde- 
pendent discussion of this case it is convenient to alter the coefficients so that the 
equation in 6 maybe in the standard form ( a , 5, c, 1) 3 =0, viz. we assume the equa- 
tion of the conic to be ( d , 3 b, 0, 0, f a , fc)£r, y, z) 2 = 0. The equation of the contact- 
locus then is 
a 2 d 2 + 4 ae 3 + 4 b 3 d-6abcd- Sb 2 c 2 =0, 
viz. this is a developable surface, or torse, of the order 4, and we at once infer 
(2*1, 1 :)=4, agreeing with (2*1, 1 : )=2m+n— 5. 
