430 Mr. W. H. L. Kusseil on the Calculus of Symbols. 

 And we consequently obtain 



Introducing the conditions 



YLiRi=o, yn_2yn-A=o, Y„_3y„-iRi=o, 



and expanding in terms of we have 



(Z'„_iZ°E-Z',_iX2Y„_3lJ._3Z'.P)2/„_i-ZtiX^Z;r2/«-2 

 -yn-i2/n-2Z;_iX^Z',P-22/„_iZtiXZ',F. 

 As the coefficient of 3/„_i in this cannot contain yn-2> we may write this 

 expression, 



Ri= (Z°_iZ«F-Z°_,X^Y„_3U„_3Z',E) -2/„-2Z°_iX^Z',E 



+(z^,z^,_,zoF-z^2z^._lX%,_3TI«_3Z',.^-~2z^,z^lXz;E)i/„_,. 



Let us now assume 



where E;^ is the mth remainder, and N„, does not contain or 2/„_„j_2' 



Hence, expanding in terms of we have 



Il.= (ZL,«L,„+2/„_„,Z',_,,L,„) 



+ (ZL,.M„,+2/«-«.Z'„_„,M,,)2/n-,«-i 



+ (ZL„.N,„ + 2/„_„.Z',_,„N,„)2/«-,. 

 =Z°_„,L„, + Zt,,M,„2/„_,,_, 



+ (ZL,„_iZ'„_,,^L,„ +Z°_,„_iZ°_,„N„,)y,,_„i. 



Now 



•— ^{U„_„i_3(Z°_„j_iZ'„_,„L,„ + Zfj_,„_iZ°_,„N,„) } 

 ax" 



= ^ { Y„_,„_3 + 2/n_,«-2Y„_,,_3}U„_,„_3 



{Z,L._xZ'„_,,L„,4-Zt,,_,Z«_,„N4 

 =X%,_,„_3lJ„_,„_3(Z.L,,_iZ'„_,,L„. + Zt_,ZL,,N^^^^ 

 + XXZ?,_,„_iZ'„_,J.„,+Z«_,,_iZll_,„NJ . 



+ 2X(Z^,„_iZ'„_,A.+ZL,„_iZo_„X02/«-,«-i 

 + (Z°_,„_iZ'„_,,L„,+ZL,„_iZ^,„N,„)2/.-,«. 

 Hence R,„+i= (Zt,,-X^Y„_,,_3lJ„_,„_3ZL,_iZ',,_,„)L„. 

 — X^Y„_„j_3U„_„j_3Zf,_,„_;^Z°_„jN„j 

 — (X-Z°_,„_iZ'„_,„L„, + X"Z°_„,_iZ°_„^N,„)?/„_,„_2 



Now consider for a moment the equations 

 L„.+x=G„L„, + H,„M„. + K„,N„, 

 M,,+i=G'„,L^+ H'^M,„+ K',,N„„ 

 N^+i= G"„L,, + H",„M,,+ K",,N,„, 



