Two methods of treatment have been carried on in parallel lines, the **unsymbolic** and the symbolic; both of these originated with Cayley, but he with Sylvester and the English school have in the main confined themselves to the former, whilst Aronhold, Clebsch, Gordan, and the continental schools have principally restricted themselves to the latter.

This enabled David Hilbert to produce a very simple **unsymbolic** proof of the same theorem.

MacMahon by **unsymbolic** methods, and later G.

Where the multiplications on the leftand right-hand sides of the equation are symbolic and **unsymbolic** respectively, provided that m P4, M P4 are quantities which satisfy the relation exp (M14+Moir+...+Mp4EpnP+...) =1+mic -Fmoif+...+mp,eng+...; where E, n are undetermined algebraic quantities.