362 ON THE DOUBLE TANOENTS OE A CUEVE OF THE FOURTH OEDEE. 
different numerical factor in the theorem, as stated in the memoir on the conic of five- 
])ointic contact, referred to above. 
If, as above, U is a quartic function y-> ^)^ 2,b^)U, then 
U, is a cubic function, and we have 
(Aj, b,, Cj, F], Gj, Hi -7-6 =2Si . TJi, 
where it is to be noticed that S, denotes a quartic function in the coefficients of Ui, and 
consequently a quartic function in (^i, y,, 2 : 1 ), the coefficients being quartic functions of 
the coefficients of U. On writing [x, 2 :) in the place of (x^, z^), Si becomes a quartic 
function of (x, y, z), which is in fact a quarticovariant quartic of U. 
If in the foregoing equation we write (x, y, z) in the place of (Xi, y^, sj, then U, 
becomes equal to 2U ; and consequently, if U = 0, the right-hand side of the equation 
vanishes. Moreover Ji, Ci, /I, 7q) (the second differential coefficients of Ui) become 
equal to («, h, c,f, g, h), and consequently the coefficients (Ai, Bi, Ci, Fi, Gi, hJ become 
equal to (a, b, c, f, g, h). Hence, assuming always that U=0, the equation becomes 
(a, b, c, p, g, hXB,, 'dy, BJ"Hi=0, 
where after the differentiations (^,, y^, z^) are replaced by (x, y, z). This is the form 
which is required for the present purpose. 
Eeturning to the foregoing expression of — 9(a", b", c", p", g", H"3[b,U, bj,U, 
this now becomes 
-9n3=-9(A", b", c", p", g", H"Xd,U, B,U, 
= 4{ 3H . (a, b, c, f, g, hXB., 'dy, c>J"H-(a, b, c, f, g, B^U, B^U)M, 
so that the equation n 3=0 gives 
ni=(A, B, c, p, G, hXB^U, Bj,U, 3H .(a, b, c, p, g, hJB^, B^,, B^)'H=0, 
and the equivalence of the equations ni=0 and n 3=0 is thus established. 
