The Dirichlet and Vibration Problems for Linear Elliptic Differential Equations of Arbitrary Order.

by prefixing 4 to each term of the formula for {1 }, k = 1; the terms beginning with 3 are obtained by prefixing 3 to each term of the sum of the formulas for {2} and { 12}, k = 1,({1}2 = {2} + {12}); the terms beginning with 2 are obtained by prefixing 2 to each term of the sum of the formulas for {3} and 121 },k = 1,(f2}l}) = {3} + {21 }); the terms beginning with 1 are obtained by prefixing 1 to each term in the sum of the formulas for {4} + {31},k = 1,({3}{1} = {4} + {31})andthetermsbeginningwith 0 are obtained by prefixing 0 to each term of the formula for {41 }, k = 1. We conclude with the remark that the master-formulas for k = 3 may be readily derived from those given above (for k = 2). Thus the terms beginning with 3 in the master-formula for {312}, k = 3, are obtained by prefixing 3 to the terms of the master-formula for { 121, k = 2; the terms beginning with 2 are obtained by prefixing 2 to the terms of the sum of the master-formulas for {21} and { 13}, k = 2; the terms beginning with I are obtained by prefixing 1 to the terms of the sum of the master-formulas for {31} and {212), k = 2; and, finally, the terms beginning with 0 are obtained by prefixing 0 to the terms of the master-formula for {312}, k = 2.