Mr. W. H. L. Russell on the Calculus of Symbols, 425 



But 



— Z'n_2Y„_2UM-2Z'n-lZ°F4- Z'^_2Y„_2U„_2Z'^_lY„_lU„_iZ'„F) > 



= Y._3U._3(Z'._2ZLiZr.F - Z'„_2Z°_, Y,_iU._iZ;F 

 -Z;_2Y„_2U,_2Z'„_iZ«F+Z;_2Y._2U,_2Z',_iY,_iU._iZ'„F) 

 +2/n-2(Z'._2Z«_iZ°F-Z',_2Z,t,Y._,U._iZ'.F 

 — Z'„_2Y„_2U„_2Z'„_iZfjF + Z'„_2Y„_2U^_2Z'„_iY„_iU„_iZ'„F) ; 

 whence we find 



Z^_2Z^_iZ°F — Z^j_2Z° _iY^_iU„_iZ',^F 

 — Z°_2Y^_2U^_2Z'^_iZ°F + Z°_2Y„_2U„_2Z'„_iY^_iU„_iZ'^F 

 — Y„_3TJ,i_3Z'„_2Z°_iZ°F + Y,j_3U„_3Z'^_2Z" _iY^_iTJ„_iZ'i^F 



+ Y._3U,_3Z',_2Y,._2U._2Z',._iZ«F 



— Y„_3U„_3Z',j_2Y„_2U„_2Z',j_iY^_iTJ„_iZ'^F. 

 Hence we infer the following rule for the formation of R^, 

 Construct the term 



Y,i_yUjj_yZ n-r+l^n-r+l^}i-r+V^ n-r+2 

 .... Y^_2XJ„_2Z'„_iY„_lU^_lZ'„F. 



In any symbol Z'„, the accent may be changed into a zero, i. e. we may at 

 pleasure substitute Z°j anywhere for Z',„ ; but in such case the previous 

 symbolical factor Y^_iU„j_i must be omitted. This term is positive or 

 negative according as the symbol Z' occurs an even or an odd number of 

 times in it ; the aggregate of all the terms thus formed constitute the re- 

 mainder R^, and the conditions that F may be externally divisible by 



d 



— are 

 dx 



V,F=0, V,_iRi=0, V,_2R2=0, V,_3R3=0, &c. 



We shall now investigate the conditions that ~ + V may externally 



divide F where P is a function of {x) and (y) . 

 As before, V^F=0, and in consequence 

 F=Z°F+3/.Z',F. 



Now 



-iZ'wF = Y^_iU„_iZ',jF 



+ PU._iZ'.E+2/nZ'.F. 

 Hence we shall have 



Ri=Z«F-Y._iU._iZ',F-PU,_iZ',F. 

 We have V^_iRi=0 in order that this remainder may contain only the 

 first power of yn-\, and 



Ri=Z»_iZ«F-Z°_iY._iU._iZ'.F-Z»_iPU,_iZ;F 

 +y«-i(Z'n-.iZS;F-Z'„,iY,.iU,_iZ',F-Z;.iPU,_xZ'„F), 



2 I 2 



