690 



\|/ being determined from (p by the equations -^ssa^+y; and the solu- 

 tion of 



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- 

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

 the factor \ and the performance (repeated m — \ times) of the 

 inverse operation (<3D)-i ; and it will be seen that, in all cases 

 where X=0, it is sufficient to perform the direct operation <pD a 

 single time. 



It is a remarkable phenomenon connected with the solutions last 

 mentioned, that they are instantaneously convertible into definite 

 integrals by changing <pTi into <pz^ multiplying by s^-^, changing 

 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"w+-=0, 



X 



ianX-\-h^Vi'^u^..^{a^x-\-h^u—(), 



and other forms. 



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

 finite differences gives solutions for the equations 



{cLnX + bn)Ux-\-n + + («i^ + ^ i )w^+ 1 + («o^ + ^0)^0?= 



(cLnX + K)^"^!, + .... + (ttiiT + 61 ) Aw^ + {a^x + hQ)u^=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 



u,=c/''(anv- + ..a,v-^a,r\\v--a)\v-(3)^\..v'^-^dv 



+y^^(«.^-+..«:^+«o)-vo(^-^>'^^<^^-^>^^-''^^ 



+ &c. ; 



and that of the second is somewhat similar. 



