and if the coefficients in the new form be represented by accenting the old coefficients, then
| a′0 = a0αn + a1nαn−1γ + ... + anγn, a′1 = a0αn−1β + a1 {(n−1) αn−2βγ + αn−1δ} + ... + anγn−1δ, · · · · · a′n = a0βn + a1nβn−1δ + ... + anδn; |
(ii.)
and this is a homogeneous linear substitution performed on the coefficients. The totality of the substitutions, (i.), for which αδ − βγ = 1, constitutes a continuous group of order 3, which is generated by the two infinitesimal transformations y(∂/∂x) and x(∂/∂y). Hence with the same limitations on α, β, γ, δ the totality of the substitutions (ii.) forms a simply isomorphic continuous group of order 3, which is generated by the two infinitesimal transformations
| a0 | ∂ | + 2a1 | ∂ | + 3a1 | ∂ | + ... + nan − 1 | ∂ | , |
| ∂a1 | ∂a2 | ∂a3 | ∂an |
and
| na1 | ∂ | + (n − 1)a2 | ∂ | + (n − 2)a3 | ∂ | + ... + au | ∂ | . |
| ∂a0 | ∂a1 | ∂a2 | ∂au−1 |
The invariants of the binary form, i.e. those functions of the coefficients which are unaltered by all homogeneous substitutions on x, y of determinant unity, are therefore identical with the functions of the coefficients which are invariant for the continuous group generated by the two infinitesimal operations last written. In other words, they are given by the common solutions of the differential equations
| a0 | ∂ƒ | + 2a1 | ∂ƒ | + 3a2 | ∂ƒ | + ... = 0, |
| ∂a1 | ∂a1 | ∂a2 |
| na1 | ∂ƒ | + (n − 1)a2 | ∂ƒ | + (n − 2)a3 | ∂ƒ | + ... = 0. |
| ∂a0 | ∂a1 | ∂a2 |