Royal Society. S75 



rations of the form ^D upon the second side X; multiplication by 

 the factor x-^ ; and the performance (repeated m — \ times) of the 

 inverse operation (^D)-'; and it will be seen that, in all cases 

 where X=0, it is sufficient to perform the direct operation ^D a 

 single time. 



It is a remarkable phenomenon connected with the solutions last 

 mentioned, that they are instantaneously convertible into definite 

 integrals by changing (pD into <pz, multiplying by e*-^, changing x~'^ 

 into D'-' (D' denoting differentiation with regard to z), and assign- 

 ing proper limits for the integral. In this manner definite integrals 

 are immediately found for 



D«M+-=0, 



X 



D»M+a?.M=0, 



(a«a; + ft«)D'*M + . . + {UfjK + io) w = 0, 



and other forms., 



6. The application of the principle above stated to equations of 

 finite differences gives solutions for the equations 



{anX-\-bn)Ugj,n+ + (aia? + &i)war+i + («o^ + ^oV^=Q*> 



and where the number of operations to be performed is denoted by 

 a fraction, solutions are found in the form of definite integrals. 

 The solution of the first when Q^=0 is 



tly 

 +y/3(a»^;'»+. •a.t' + ao)" V''(^-'^)M^-^)^2 • • -^^-^^^ 



+ &c. ; 



and that of the second is somewhat similar. 



From some investigations effected by interchanging the symbols 

 X and D in the solution of the general linear equation in finite dif- 

 ferences of the first order, it would seem that definite summations 

 may be used to represent the solutions of certain forms of equations. 

 Thus a partial solution of 



is c2(r;2!)»f*« from «= — a to z==0. 



7. In attempting the solution of some equations by means of suc- 

 cessive operations, not consisting exclusively of D combined witli 



