688 



so written that the operations included under the function ^ are not 

 performed or suppressed, then (p(T>,—x).u=X has for its solution 

 u=^(D,—x},X" The solution thus obtained may not be, and often 

 is not, interpretable, at least in finite terms ; but if by any trans- 

 formation a meaning can be attached to this form, it will be found 

 to represent a true result. 



An important solution immediately deducible from this principle 

 is given by Mr. Boole in the Philosophical Magazine for February 

 184^7, and is extensively employed in the present paper. It is imme- 

 diately obtained by making the conversion above proposed in the 

 general equation of the first order and its solution. 



2. By the use of this theorem and the general theorems above 

 referred to, the solution of the equation 



is found in the form 



^^=^"^£-/Q^'i•(D2 + c2)'«-l{a?-l(D2-fc2)-'"(^c-(»^-l).fW•^.P)} ; 



of which various particular cases and transformations are given and 

 discussed ; including the well-known forms 



2 m 



X — 



and extensions of these forms. 



The application of the process to equations of the third and higher 

 orders gives rise to solutions of analogous forms ; and in particular 

 the equation 



(«„a?+^JD«M4- ... -i-(t«ia? + 6i)Dw + («o« + Jo>=X 

 is solved in the form 



^=(«A+ ..+«iD + e^o)-^£««''(D~a)^(D-/3)^.. 



(^-^{s4?(D-a)-^(D-/3)-B,..x}), 



where zr—- . , -r h h + 



and by the application of the theorems first referred to, a still more 

 general form is solved. 



The solutions above-mentioned are subject to the important re- 

 striction that m, A, B, &c. (denoting the number of times that the 



