In the Proceedings of the Edinburgh Mathematical Society, 1948, there appear two papers by Lars Gårding and Turnbull respectively (Gårding [1], Turnbull [2]) which formulate the theory of Cayley and Capelli operators associated with symmetric matrices. Turnbull derives the modification, appropriate to symmetric matrices, of Capelli's Theorem, which states that (taking a third order operator for the sake of ease in writing)

where the symbol (xyz)ijk stands for the determinant

with a similar meaning for the determinantal differential operator, while the symbols are polarisations (Capelli [1]; cf. Turnbull [1], p. 116). Gårding's theorem deals with the effect of such modified Capelli operators on powers of the determinant of the symmetric matrix in question. The subject of this note is an alternative derivation of the modified Capelli theorem and of Gårding's theorem.