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ployed, and consequently the equation is reduced an order lower ; 

 it is therefore integrated when of the second order. There is a se- 

 ries of equations of this class, each essentially distinct from the rest, 

 yet all reducible by a similar process. 



These equations contain two arbitrary functions of x. The num- 

 ber therefore of particular practicable forms, which may be deduced 

 from each, is very great, a circumstance which renders our chance 

 of putting any proposed equation under one of these forms greater 

 in the same proportion. On account of the very large number of 

 particular integrable equations which each general example furnishes, 

 selection would be very difficult, and all could not be given ; the au- 

 thor has therefore refrained from giving any. 



The second class of equations may be deduced from the first by 

 the interchange of the symbols D and x, and changing r into f~^. 

 The second general theorem can be deduced from the first in like 

 manner; and this class may be transformed and reduced by it in a 

 manner exactly similar to that by which the former class is reduced 

 by the first general theorem. The solution therefore of the one se- 

 ries may be deduced from that of the other by the interchange of 

 symbols oiily. But in the second series the solutions obtained are 

 not always practicable, that is to say, they cannot always be inter- 

 preted in finite terms. They have therefore been reduced by the 

 introduction of new arbitrary functions of D, which render them 

 practicable ; this process however necessarily diminishes their gene- 

 rality. 



When reduced to the ordinary form, these equations arc somewhat 

 complicated ; but by giving suitable forms to the arbitrary functions 

 of D which they contain, we may derive from them particular ex- 

 amples of a form as simple as we please, and by introducing as many 

 arbitrary constants as possible, these examples may be made very 

 general of the class to which they belong. In the integration of 

 linear equations, the coefficients of which are integer functions of x, 

 they may prove very useful. 



Next, an equation, a particular case of which was treated by Mr. 

 Boole in the Cambridge Mathematical Journal, is here integrated 

 under its most general form. Instead of integer functions of the 

 coefficients may be any functions whatever, consistent with the con- 

 dition of integrability, which is ascertained, and the formulae of re- 

 duction assumed by Mr. Boole are shown to be universally true. 

 An additional function of the independent variable is also introduced 

 into the operating symbol tt. The equation therefore, independently 

 of the condition of integrability, contains two arbitrary functions of 

 X, and consequently gives rise to a considerable number of particular 

 integrable examples. 



Here also the interchange of the symbols D and x is made, both 

 in the equation to be integrated and in the general symbolical 

 theorem by which it is reduced, and the same reduction to prac- 

 ticable forms as before is likewise made. 



The next class of equations results from the generalization of an- 

 other equation integrated by Mr. Boole in the Cambridge Mathema- 



