DP1DP2  ... Dy, . (lal) (i*2) ... (Pm) = A, and the

law by which the operation is performed upon the product shows that the solutions of the given problem are enumerated by the number A, and that the process of operation actually represents each solution . Ex . Gr.—Take a1=3, X2=2, 711=I, p1=2, P2=2, P3=11 P4=I, D z D 1a3a2a1= 8, and the process yields the eight diagrams: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 ~1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 viz. every solution of the problem . Observe that transposition of the diagrams furnishes a proof of the simplest of the laws of symmetry in the theory of symmetric functions . For the next example we have a similar problem, but no restriction is placed upon the magnitude of the numbers which may appear in the compartments . The function is now halhr2... ham, h„. being the homogeneous product sum of the quantities a, of order X . The operator is as before DP1Dp2 ... Dp ...