426 



Mr. W. H. L. Russell on the Calculus of Symbols, 



since 



Y„_,U.-2Z'._.oZ«F-Y„_2U„_3Z',_iX,_,U„_,Z'„r- 

 Y„_2U,,_2Z'„_iPU,_iZ',,F + y„_i(Z'„_iZ»F 

 -Z'„_iY._iU„_,Z'„F - Z'„_iPU„_,Z'„F) + 

 PU,,_,Z'._,Z«F-PU,,_,Z'„_,Y„_,U,._iZ'„F 

 ~PU,_2Z'„_iPU„_iZ'„F. 



Whence we find that + P divides Ri with a remainder, 



dx 



R3 = Zr^Z^F - Z°_,Y,,_,U„_iZ'„F 

 -ZLiPU„_iZ'„F-Y„_3U«-2Z'„_iZ°F 

 • + Y„_2XI,_2Z'„_,Y,_,U„_iZ',J + Y„_ A.-2Z'„_iPU„_,Z',,F 



- PU,_2Z',_iZ?,F + PU,_,Z'„_iY„_iU„_iZ'„F 

 + PU,_2Z',_iPU„_iZ',F. 



Putting V,"i_2R2 = 0, we find in like manner, 

 Il3=Z«_2Z°_iZ?,F-Zt2Z»_iY„_,U,_iZ',.F 

 -Z°_2ZLxPU,._iZ'„F-Z°_2Y,,_2U„_2Z'„_,Z«F 

 + Z^2Y„_2U,_2Z',_iY._iU„_iZ',F + Y„_2U,_2Z'„_iPU„_iZ',.F 

 -Z°_2PU„_2Z'„_XF+Z°_2PF„_2Z'„_iY,_iU„_iZ;F 

 +Z^2PU.-2Z'„_iPU„_,Z',J-Y„_3U._3Z'„_2ZtxZ?,F 



+ Y„_3U,._3Z',._2Z «-iY„_iU„_iZ'„F 



+ Y„_3U„_3Z'„_2Z^;_iPU,„_iZ'„F 



4" Y„_3U,j_3Z ,i_2Y,j_2U,j_2Z ,j_iZ^jF 



-Y,_3lJ,,_3Z'„_2Y„_2U,,_2Z',_iY„_,LT,_,Z'„T> 



-Y„_3U,,_3Z',_2Y„_2U„_2Z'„_,PU„_,Z'.F 

 + Y,._3U„_3Z'«_2P„_2U„_2Z',,_,Z«F 



-Y,_3U„_3Z',_2PU„_2Z',,_,Y,_iU„_iZ'„F 

 -Y,_3U,_3Z'„_2PU,„_2Z'„_iPU„_iZ',F 



- PU._3Z V2Z?._iZ°F + PU,_3Z'„_2Z«_iY,_iU,_,Z',,F 



+ PU„_3Z',_2Z,LiPU,_iZ'„F + PU„_3Z',_2Y„_2U„_2Z'„_iZl',F 

 -PU„_3Z',_2Y,_2U,_2Z',,.iY,_iU„_iZ;F 

 ~ PU„_3Z;_2Y„_2U,_2Z'„_iPU,,_iZ',F 

 + PU„_3Z;_2PU,_2Z'„_iZ«F 



-Pu,,_3z;_2Pu,_2Z'._,Y,._,u,._,z;r 



-PU._3Z;_2PU _2Z;_iPU„_iZ',.F. 



We see at once that the value of R^ in this case can be formed from that 

 calculated in the last example, by writing P at pleasure for any one or 



