Mr. W. H. L. Russell on the Calculus of Symbols. 437 



We may consider B=l, which gives the following conditions: a^b=ab', 

 a=b'. Also 



A=?^, C=L B='-. F=£. 

 a b' a a 



And the symbols may he written 



b(ax + by + c)- a(aa? + Jy + c')— , 



and 



«V + 2ab xy + 5y + 2ac'a? + 2cby + H. 

 It follows hence that in order to find the form of .the differential equa- 

 tions to which these symbols give rise, we must know the expansion of 



(d d\'"' 

 +Y^J , where X and Y are functions of a? and y. 



The expanded form will be a series of terms like 



uiX Tx-V ("y-Y U-\ 



' \ dy) \ dx) \ dyj \ dx) 



(d \^ d 

 X in powers of It 



must be remembered that X is a function of x and y, in which (y) during 

 the present process is considered as constant, and therefore X may be looked 

 upon as a function of {x) only. 



Now we shall find after a few differentiations, that 



5 J5 



Now let 



dx' 



+(xax^+x2axHX3axHx^^x) — , 

 + (x^xax^ + xax^ax^ + x^axax^ 

 + xax^ax+x^ax^ax+xTOX)^ 



+ (xaxaxax2+ xaxax^ax+xgx^axax) ^ 



+ xaxaxaxsx ~, 



aw 



Then 



-7* 



i_ 



xw=s^x«ax^axy ... 



where there are r a's, and a4-^+7+....=w. Hence we shall have 



