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705 
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y find 
Since (2) is the differential with respect to t of (1), the result of eliminating t 
between these two equations is the discriminant of (1). Hence the equation of the 
required surface is the discriminant of (1) with respect to t. Since (1) is only of 
the fourth degree, this discriminant is easily formed. If (1) be written in the form 
At* + 4 Bt 3 + 6Ct 2 + 4>Dt + E = 0, 
it will be found that A and B do not contain x, y, z, while C, D, E contain them, 
each in the second degree. Now the discriminant is of the sixth degree in the 
coefficients, and of the form Acfr + B^ (see Salmon’s Higher Algebra, § 107); hence 
it contains x, y, z only in the tenth degree. This is therefore the degree of the 
required surface. 
The section of this derived surface by the principal plane z consists of the dis 
criminant of 
2-1 2-1 
a'? b 
(which is of the sixth degree, and is the first negative pedal of —= together 
with the conic (taken twice), which is obtained by putting t = 2c 2 in (3). 
This conic, which is a double curve on the surface, touches the curve of the 
sixth degree in four points. 
2. The formulae for the conic are quite analogous to those for the ellipsoid, viz. 
we have 
x = X {2 -1 (X* + r*)J. y = Y ja - 1 (X* + y*)}, 
leading to the equations 
and its derived equation, from which to eliminate 6. The first is the cubic equation 
(A, B, G, B){6, l) 3 = 0, where 
A = 1, B- — |(a 2 + b 2 ), G = %(a 2 x 2 + b 2 y 2 + 4 a 2 b 2 ), D — — 2a 2 b 2 (x 2 + y 2 ). 
C. X. 73