424 Mr. W. H. L. Uussell on the Calculus of Symbols. 



Having thus explained tlie notation I propose to make use of, I proceed 

 to determine the conditions that F may he externally divisible by or, 



in other words, that F may be a perfect differential with respect to (x). 

 It will be seen that the above notation will enable us to obtain expressions 

 for the conditions indicated by the process of M. Sarrus. 



It is obvious that if we expand F in terms of y^f in order that the sym- 

 bolical division with reference may be possible, the terms involving 



ax 



Vni Vni &c. must vanish. 



Hence V„^F=0, and consequently 



where, of course, Z^F, Z'^F do not contain 

 Hence we have 



^(U„.,Z'.F)=Y,_iU„_iZ^F+y.Z^F, 



and therefore F becomes 



-^(U,_iZ'„F) + Z,T- Y,_iU„_iZ'„F ; 



and if Ri be the first remainder, 



Ri=Z,»F-Y,_iU,_iZ',F. 



The condition that this may be divisible by will be, as before, 



Y^_illi=0 ; hence Ri becomes 



Z°_iZ«F-Z«_iY,_iU._iZ'.F+2/.-i(Z'i-nZ°F-Z'._iY,,_iU,_iZ'.F). 



Now 



^U„_2(Z'„_iZ^F — Z'„_iY^_iU^_iZ'„F) = 



Yn-2U„_2Z'„_lZ^F— Y„_3U„_2Z'„_iY„_lU„_lZ'„F 



+2/.-i(Z'._iZ»F-Z'„_iY,,_,U._,Z'.F) ; 

 and if R2 be the second remainder, we find 



R2 = Z^_iZ°F — Z° _i Y^_iU„_iZ'„F 



— Y^_2U„_2Z'^_iZ°F+Y„_2U^_2Z',i_iY„_iU„_iZ'„F : 



the next condition is V^_2R2=0, and therefore 



R2=ZL2zriZ°F 



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



+ Zw_2Y„_2TJ,i_2Z'ji_iY„_iU^_iZ'„F 



+2/n-2(Z;_2Z»_iZ»F-Z'._2ZriY,_iU._iZ;F 



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



