Mr. W. H. L. Russell on the Calculus of Symbols. 429 



Let us now assume 



Then is formed according to the following rule : — Form the term 



.... XT„_3"U._3Z'„-iXX,_2U„_2Z'.F. 



Z'„j may in any place be changed into Zl, ; but in this case either the 

 preceding XY„j_2U„,_2 must be omitted, or the succeeding XY„j_iU^_i 

 changed into X. The signs of the terms follow this law. A term not 

 containing X introduced in place of XYU is positive if Z' occurs in it an 

 even number of times, negative in the contrary case. But every X in- 

 troduced in place of XYU occasions a change of sign. The aggregate 

 of all the terms thus formed will give M^. 



We form N thus : construct the term 



XZ',j_^^yXY„_rU„_rZ',j_^^2XY,i_,.^iU,j_,.^i Z'^F, 



and a precisely similar rule holds good. is subject to the condition 



VL A= 0, v„_,_iV,_ A= 0. 



Let us now investigate the criterion that F may be divisible by 



where P and Q are functions of (^x) and (y). 

 Proceeding as before, we have 



+ ^) (Y.-2+yn-iV._2)U,._2Z'.F+QU,_2Z',.F 



= (XY,_2+ PY,,_2+ Q)U,._2Z„F +2/,._i(X + P)Z'.F+y.Z',F. 

 The form of this equation gives us the following rule to ascertain the 

 successive remainders. Construct the remainder in the last case as before, 

 and substitute at pleasure Q in any place where XY is found, P in any 

 place where X is found. The aggregate of the term thus formed will give 

 the remainder in this case. 



We now investigate the condition that — ^ may be an external factor 



(too 



of F. 



We put, as before, F=Z°F + 2/»Z',,F, where Z'„F must contain neither 

 y^-x nor 2/n-2> which gives the conditions 



Now we have 



^3 (tJn-3Z'.F) = ^(Y,._3 + 2/n-2Yn-3)U«-3Z'.F 

 = ^(Y._3U._3Z'.F + y.-2Z'nF) 



=X^Y._3TJ._3Z'nF+t/„_2X^Z',,F4-22/._iXZ',F+2/nZ'nF. 



