WHICH SATISFY GIVEN CONDITIONS. 
131 
where (a, <p) are arbitrary. In fact these values give 
^A = — Jclc i cos 2 <p — kJc x sin 2 <p, 
B = — Jc(2a,Jc x -\-V) cos 2 (p — Jc x (2a&+l) sin 2 <p, 
C = — a^ + 2) cos 2 <J5 — Jc x a{ ctk-\-2) sin 2 <p, 
= — ka 2 cos 2 (p — Tc x a 2 sin 2 <p , 
whence u being arbitrary, we have 
i(A, B, C, DX*, l) 3 
= — \Jc cos 2 <p (Jc x w-\-\')-\-Jc x sin 2 <p (kw -j- l)](<w + a) 2 , 
viz. the equation (A, B, C, DX*>, 1) 3 =0, considered as a cubic equation in a, has the 
twofold root a— —a, that is, we have the above relation between (A, B, C, D). Whence 
also writing sin<p = j-^^, cos <p = |~^ 2 , the equation of the surface is satisfied by the 
values 
x-\-y : x— y : z : w=.\J (1 — 7g«)2a(14-A 2 ) 
: v/fd+fe (1-^) 
: (l+X? 
: (2._i)a-x7+(2«- 5)^. 
or the coordinates are expressed rationally in terms of a, A. 
Annex No. 4 (referred to, Nos. 22 and 71). — On the Conics which touch a cuspidal cubic. 
In the cuspidal cubic, if #=0 be the equation of the tangent at the cusp, y= 0 that 
of the line joining the cusp with the inflexion, and 2=0 that of the tangent at the cusp, 
then the equation of the curve is y 3 =x 2 z ; the coordinates of a point on the cubic are 
given by x : y : 2=1 : 9 : 9 3 , where $ is a variable parameter; and we have, at the cusp 
9=co , at the inflexion 9=0. In the cubic, m—n— 3, a(=3?i+«) = 10. 
Considering now the conic 
' (a, b, c,f, g, hjx, y, zf= 0, 
this meets the cubic in the 6 points the parameters of which are determined by the 
equation 
(a, b, c,f i g, hj 1, 9, 9 3 ) 2 =0, 
or, what is the same thing, 
(c, 0, 2 f, 2 g, b, 2 h, 1) 6 =0. 
The discriminant of this sextic function contains the factor c, hence equating the 
residual factor to zero, we obtain the equation of the contact-locus in the form 
(c,f, g, b , h, a) 9 = 0. 
u 
MDCCCLXVIII. 
