For Noether's second theorem, we consider those variational symmetries (called gauge symmetries) which are parametrized linearly by a set of arbitrary functions and their derivatives. These variations have the generic form δ δ λ λ u σ σ = R a σ σ λ λ a + R a σ σ , i λ λ i a + ⋯ ⋯ + R a σ σ , i 1 . . . i s λ λ i 1 . . . i s a = ∑ ∑ | I | = 0 s R a σ σ , I λ λ I a , where the coefficients R a σ σ , I can depend on the independent and dependent variables as well as the derivatives of the latter up to some finite order, the λ λ a = λ λ a ( x ) are arbitrarily specifiable functions of the independent variables, and the latin indices a , b , … … take the values 1 , … … , q , where q is some positive integer.