939 



tical Journal. Here the symbol D of Mr. Boole is replaced by the 

 general symbol ro", and moreover the first member of each equation 

 contains two arbitrary functions of ra- ; and by means of another 

 extension, this example gives rise to a whole series of equations con- 

 stituting a class. The reduction is effected partly by the first general 

 theorem in the calculus of operations, and partly by other means. 

 It must be observed that each of the classes is totally distinct from 

 the others, and its mode of treatment also distinct; also each of the 

 general examples in the series contains two arbitrary functions of the 

 independent variable, and will therefore give the solutions of a large 

 number of particular equations, but for the reason before stated 

 particular examples are not given. 



Here likewise, by the interchange of the symbols D and x, another 

 series of equations with their solutions or reductions is obtained, and 

 also another general theorem by which they may be transformed and 

 reduced. But the solutions of the examples of the one series may 

 be deduced from those of the other by the interchange of symbols. 

 It is not a little remarkable that this interchange of symbols in all 

 these cases should be found possible, it will however be found pos- 

 sible in another case to be hereafter described. 



The last class of equations discussed in this paper is transformed 

 by means of a general theorem of a very different kind from any of 

 those which have been employed in reducing and integrating any of 

 the previous classes. By means of this transformation, the symbol 

 ^3", of which the first mernber of these equations is a function, is 

 placed in a position to operate upon the whole of that member, a 

 certain equation of condition among the coetficients being previously 

 admitted. Hence by operating upon both members M'ith the inverse 

 of this symbol, the equation is once integrated, and, if it be of the 

 second order only, completely solved. 



Here too the interchange of symbols may be made both in the 

 equation and its solution, and the solution so changed will be the 

 solution of the equation changed in like manner. The general sym- 

 bolical theorems, which here consist of a series of terms, may be de- 

 rived the one from the ather in the same way, and by changing the 

 signs of the alternate terms. 



Reductions of the arbitrary functions of D, similar to those before 

 made, are made here also ; and by particularizing some of the func- 

 tions so reduced for the sake of simplification, several very singular 

 resulting equations are obtained. If in these we assign to the re- 

 maining arbitrary functions, particular forms, and introduce as many 

 arbitrary constants as we can, we may find particular examples 

 which may be of great use in the integration of equations with 

 coefficients containing only integer functions of x. 



By a very obvious substitution an arbitrary function of x may be 

 introduced into any of this kind of equations, and also another func- 

 tion of D, and the last often with great advantage. 



2. " On the Oils produced by the action of Sulphuric Acid upon 

 various classes of Vegetables." By John Stenhouse, Esq., F.R.S. 



