E,rpressinci Solutions of Laplace's Eguation. 205 



Using the extended symbolical i'orni of Leibnitz^s theorem, 

 we have 



fmuv = vf{J))v + -Du.^f{D).v+'^u.^JCD).v+... (40) 



where J) = (l/d.v. 



Putin this D= f, 

 a/ji 



an d rem ember that, in differentiation with regard to D, 

 vl — fi^ is to be treated as a constant. 

 We have then 



=/.j„. (v/i=^^ -)/.«-+ v/r-7^.j,„'(yr^|^)^»-^. (41) 



But w^e have 



J./ = i(Jm-i-J.+i) (42) 



Let me denote n{J^ ?:(/.) by P^^. 

 Then 



p:=j„^(./r:i7^A)/^^ .... (43) 



and the equation above gives 



PT = /.pr,-i + ^\7 ^' {K-i - Pr!) , . . (44) 



which is one of the recurrence formulae. 

 But we have also 



•VW = ';^-J,„W-J»-.iW (15) 



