AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 187 
The invariants will in this, and in several of the examples which follow, be calcu¬ 
lated from the results given in Salmon’s ‘ Higher Plane Curves,’ Second Edition, 
Arts. 217-224. 
The invariant S = mafi-yz 2 — to 4 . 
The invariant T = a 2 /3 3 y 2 z 4 — 20 m ?J afiyz~ — 8to 6 . 
Hence A = T 2 + 64S 3 = ctfiyz- (afiyz 2 -{- 8 w 3 ) 3 . 
(B.) The Conic Node Locus is z = 0. 
•r 
Transforming the equation to the new origin a, b, 0 , the lowest terms are of the 
second degree. 
Hence the new origin is a conic node on the surface. 
Hence z = 0 is the conic node locus. 
Hence A contains z 2 as a factor. 
(C.) Three non-consecutive Surfaces of the System touch each of the planes 
z = dz ( — 8 m 3 / a/3y) m at each point. 
To prove this, the tangent planes to the surfaces which are parallel to the plane 
2=0 will be found. 
The tangent plane to the surface 
a (x — a) 3 + ft (y — bf + Gto (x —a) (y — h) -f yz~ = 0 
at £ y, £ is 
(* — f)[ 3 “(f — «)' ! + 6m (v — 0] + (y — v)l*fHv — + 6m (f —«)] 
+ (z - Q 2 7 C = 0. 
If it be parallel to z = 0 , the coefficients of x and y must vanish, but the coefficient 
of z must not vanish. 
Therefore, 
a (g — af + 2 to (y — b) = 0.( 61 ), 
ft (y — bf + 2 to (£ —a) = 0.(62). 
From these, and from the condition that y, £ lies on the surface, 
2to (i — a) (y — b) + y£ 2 = 0.(S3). 
If these be satisfied, and £ do not vanish, the tangent plane is z = £. 
The solutions of (61), (62), (63) are 
?-a= 0, y — b ■= 0 , £=0. ..... (64), 
£■ — a = — 2ma -2/3 /3“ 1/3 , y — 6 = — 2ma~ 1/3 /3 -2/3 , £ = ffi (— 8TO 3 /a/3y) l/2 (So), 
£ — a = — 2mcua" 2/3 j8 _1/3 , y — b = — 2ma> 2 a -]/3 /3~ 2/3 , £ = ffi ( — 8TO 3 /a / 8y) 1/2 (66), 
£ — Cl = — 2m<u 2 a -2 3 /3“ 1/3 , ^ — b = — 2TOwa _1/3 /3~ 2/3 , £ = d; (— 8TO 3 /a/3y) 1/2 (67), 
where cu is an imaginary cube root of unity. 
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