Royal Society, 373 



1847, 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 



D'M + 2Q.DM + ^c^ + Q2 + Q'— ^ ^^ ^ ju = P, 



is found in the form 



7,f=x'»£-/Q'^(D24-c2)»'-i{a;->(D2+c2)-'»(a;-('»-i) .g/Q*'.P)} ; 



of which various particular cases and transformations are given and 

 discussed ; including the well-known forms 



D^M + — Dm + c": M = P, 

 X — 



^ ,^ / m(m—\)\ ^ 



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 



is solved in the form 



bnZ^ + bn-iZ"-^ + ..._b„ A , B 



where — s—. „_. ,- — t 1 « + ... 



a»2;»-f-a„_iz" '-)-... a„ x—a x—p ; 



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 

 operations are to be repeated) must be integer ; but in the subse- 

 quent part of the paper, a mode is suggested of instantaneously con- 

 verting these solutions into definite integrals not affected by the re- 

 striction. 



3. The interchange of symbols above suggested frequently renders 

 available forms of solution which otherwise would not be interpret- 

 able in finite terms. The ojjeration (fD)^ is not intelligible if m 

 be a fraction ; but if by any legitimate process this be changed into 

 the factor (<p{—x))% the restriction ceases to operate. By the ap- 



