406] 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
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viz. the equation {A, B, G, Dffw, 1) 3 = 0, considered as a cubic equation in w, has the 
twofold root <u = —a, that is, we have the above relation between (A, B, G, D). Whence 
i •, ■ • . 2\ 1 — A, 2 
also writing sin <p = j-—— , cos <£ = ^ , the equation of the surface is satisfied by 
the values 
(1+A 2 ) 2 
or the coordinates are expressed rationally in terms of a, X. 
Annex No. 4 (referred to, Nos. 22 and 71).—On the Gonics which touch a cuspidal cubic. 
In the cuspidal cubic, if x = 0 be the equation of the tangent at the cusp, y = 0 
that of the line joining the cusp with the inflexion, and z — 0 that of the tangent at 
the cusp, then the equation of the curve is y l = x-z; the coordinates of a point on the 
cubic are given by x : y : z — 1 : 6 : 0 s , where 6 is a variable parameter; and we have, 
at the cusp 6 = oo , at the inflexion 0 = 0. In the cubic, m=n = 3, a (= Sn + k) = 10. 
Considering now the conic 
(a, b, c, f g, h\x, y, zf = 0, 
this meets the cubic in the 6 points the parameters of which are determined by the 
equation 
(a, b, c, /, g, h\ 1, 0, 0 3 ) 2 = 0, 
or, what is the same thing, 
(c, 0, 2/, 2g, b, 2h, a$0, 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, /, g, b, h, of = 0. 
It follows that the number of the conics (1 ::) is = 9, which agrees with the general 
value (1 ::) = 2m + n. If the conic pass through the cusp we have c=0, and the equation 
in 0 is reduced to a quartic; it is convenient to alter the letters in such wise that 
the quartic equation may be obtained in the standard form (a, b, c, d, e]£0, 1) 4 = 0; 
viz. this will be the case if the equation of the conic is taken to be 
(e, 6c, 0, \a, 2b, 2d\x, y, zf = 0, 
C. VI. 
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