

874 'Royal Society. 



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

 tained for (b being integer), 



(a;2+c2)D2M— 2wDM+ft(2a— 6+ 1)m=P, 



d^u_ b{b + \) 



dt^ cosH ""• ' 



<px,'D'^u + ^x.Du + ('<p'x-^0''x)u='P. 



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 geyierally, 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 

 diflPerential equations. Thus, the equation 



d^u In d^u 

 dx'^ X dxdy 



which may be solved by m successive substitutions, receives its 

 solution in the general form 



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

 formed upon "i/ix^y) in order to arrive at the result. It v/ill be ob- 

 served that the process e*^'^' performed upon ■i^y denotes •i^{y-\-<^x). 

 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)_ 

 d^+^q[d^ + d^J+ (p+qy =?'(/''^) 



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



j^_^m-«CD2_D'2)™-i|a:-'(D'^— D''^)-™{a?«-'"+i.\J/(a;,?/)}| ; 



t|/ being determined from <p by the equations -^jcic 4:^; and the solu- 

 tion of 



(a„x+bn)^^ + (an-ix+b„-0^-^^-^+.. + (a,x^bo)~=,p(x,y) 



which is readily deduced from the solution of the corresponding 

 form in ordinary equations. 



5. The character of most of the solutions may be described as 

 follows : they consist in the performance (repeated m times) of ope- 



