- Referred to the asymptotes as axes the general equation becomes
**xy**2 obviously the axes are oblique in the general hyperbola and rectangular in the rectangular hyperbola. - If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x, 2, z 2, yz, zx,
**xy**from the six equations u = v = w = o = oy = = 0; if we apply the same process :to thesedz equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails. - Now f = (
**xy**1 - x i y) (**xy**2 - x 2 y) ... - Every symbolic product, involving several sets of cogredient variables, can be exhibited as a sum of terms, each of which is a polar multiplied by a product of powers of the determinant factors (
**xy**), (xz), (yz),... - Since, If F = An, 4) = By, 1 = I (Df A4) Of A?) Ab A"'^1Bz 1=, (F, Mn Ax I Ax 2 Axe Ax1) J The First Transvectant Differs But By A Numerical Factor From The Jacobian Or Functional Determinant, Of The Two Forms. We Can Find An Expression For The First Transvectant Of (F, ï¿½) 1 Over Another Form Cp. For (M N)(F,4)), =Nf.4Y Mfy.4), And F,4, F 5.4)= (Axby A Y B X) A X B X 1= (
**Xy**)(F,4))1; (F,Ct)1=F5.D' 7,(**Xy**)(F4)1.

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