689 



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 operation (^D)"^ is not intelligible if m 

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

 the factor (^(— a?))*", the restriction ceases to operate. By the ap- 

 plication of this principle, solutions of a simple character are ob- 

 tained for {h being integer), 



+ c2)D% — 2g!^Dm + 6(2a — 5 -f- 1 )m = P, 



dHL_ h{h-^\) 



dt^ cosH 



^-b(b-h2) , ^ , .t^ = P, 

 dt^ ^ ^(l-r-)2 



(px.I)^u + ^x.Du-]-(^'x—0"x)u=F. 



4. The advantages of the forms above given in this particular, 

 that the number and order of the operations in the solution are ex- 

 pressed generally, and not by a series of substitutions involving 

 changes of the variable as in the ordinary mode of solving Riccati s 

 equation, appear more clearly in the application to partial linear 

 differential equations. Thus, the equation 



dx^'^lcdxdy'^ \3c^ )dy^~x^-dy~~ ^2 tl—^^{x,7/), 



which may be solved by m successive substitutions, receives its 

 solution in the general form 



{aJ-l(D2— ;^^2D'2)-»^|^-(m-l)gnlog<rD';^(^,^)J | . 



which exhibits at a glance all the successive processes to be per- 

 formed upon ^(x,2/) in order to arrive at the result. It will be ob- 

 served that the process ^^^'^ performed upon denotes 

 Among other results worthy of notice on this branch of the subject 

 may be noticed the solution of 



d^u a /du du\ a(a — l)—m(m — l) 

 dpdq '^p + g\dp ^dq)'^ {p + qf ~~ ^ ^^^'^^ 



(solved by Euler in a series when there is no second term) ; viz. 



w=a;'»-°'('D2-D'2)"^-i |a-i(D2-D'2)-"'{.r«-'« + i.4/(.r.^)} j ; 



