Partial Differential Operators. 51 



in applying the symbolical form of Taylor's theorem to the ex- 

 pansion which, in itself symbolical, contains the generalization 

 of Leibnitz's theorem, thus giving rise to a symbolism of the 

 second order, a phenomenon which, it is believed, here for the first 

 time makes its appearance in analysis. 



Let (f>i,(l>2,<f>3) • • • <f> r be any functions of x,y t z... ; h 9i h y} h a ... 

 capable of being developed in a series of integer powers of the 

 latter set of variables, where it is of course understood that 



s d z d x d 



in like manner let 



w j* w d £. a 



*- dsj 6y "aX ^ ""^: ,,,, 



so that in fact h' Xi S' y , &' z , ... are abbreviated expressions for 

 S^, $dy, &d z , ... or, if we please so to say, for 



d d d 



■, d 1 d 7 d 



a -j- d^- d-r 



ax dy dz 



Let B x>i ; 8'^i signify the operants 8 X ; 8 f x restricted to operate 

 exclusively on <£ t -; finally, let 



A f ,y = ft*, i . B ttJ + S' yj i.8y,f+ S'g, i . h z ,j + , . . ; 



then giving to i, j all possible values subject io the inequalities 

 i <j, j < n+l, the following equation is true, 



$1*04*$B* • • • <f>n*= \j? A hi <t>\4>^3 • • • <£»]*• 



What follows within inverted commas isfromMr.Cayley's pen. 

 "Write 



viz. A, any function of degrees a } a ; and so 



and 



but all suffixes are to be ultimately rejected. Then 



=e %+^ l( ^ ?a )i(j w ^)?(^ ^, ^ 



= eA 01 B 21 A 10 * if Aoi^fi-f^i- 



E2 



