Sentence Examples


  • We can prove that if the three equations be satisfied by a system of values of the variable, the same system will also satisfy the Jacobian or functional determinant.
  • For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=�.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.
  • 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, �) 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.
  • Solving the equation by the Ordinary Theory Of Linear Partial Differential Equations, We Obtain P Q 1 Independent Solutions, Of Which P Appertain To S2Au = 0, Q To 12 B U =0; The Remaining One Is Ab =Aobl A 1 Bo, The Leading Coefficient Of The Jacobian Of The Two Forms. This Constitutes An Algebraically Complete System, And, In Terms Of Its Members, All Seminvariants Can Be Rationally Expressed.
  • 1 Ab' Establishing The Ground Forms Of Degrees Order (I, O; I), (O, I; I), (I, I; O), Viz: The Linear Forms Themselves And Their Jacobian J Ab.
 

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