24 The Rev. S. Earnshaw on the Transformation 



silica dissolved in the solution of carbonate of soda is obtained 

 by supersaturating the solution with muriatic acid, evaporating 

 to dryness, and treating the residue with water; the silica 

 then remains undissolved. 



A solution of caustic potash cannot be used for separating 

 the silica from the sand in the analyses of the ashes of vege- 

 table substances, but carbonate of soda must be made use of 

 for this purpose; for the sand is not in most cases pure sand, 

 but usually consists of a mixture of sand and clay, which is 

 either derived from the soil upon which the plants grew, or 

 from the floor of the barn upon which the corn was thrashed, 

 and which has been driven into the straw and the grains of 

 seed during the process of thrashing. This clay is readily 

 decomposed by a solution of caustic potash, the alumina being 

 dissolved, and the amount of silica contained in the substance 

 is rendered incorrect by the silica dissolved out of the clay. 

 When carbonate of soda is used, this decomposition of the 

 clay does not occur. 



[To be continued.] 



II. On the Transformation of Linear Partial Differential 

 Equations with constant Coefficients to Fundamental Forms. 

 By the Rev. S. Earnshaw, M.A.* 



THERE are to be met with in books occasional instances 

 of the transformation of equations of the class here pro- 

 posed to be considered, but I am not aware that the subject 

 has ever been entered upon systematically. A iesiv years ago, 

 when engaged in some speculations on Laplace's equation, I 

 effected the transformation of all partial differential equations, 

 with constant coefficients, of the second order, for two and 

 three independent variables. The method seems applicable 

 to higher orders and any number of variables. The result of 

 my investigations is, that every partial differential equation of 

 the second order, with constant coefficients, can be transformed 

 into one or other of these two forms, when there are two in- 

 dependent variables, 



— - 



dxdy ~ ^ '' 



d^u du 



d^^^'^Ty' <^-) 



and into one or other of these two forms, when there are three 

 independent variables, 



♦ Communicated by the Author. 



