Calculus of Operations to Algebraical Expansions. 353 



the algebraical law of indices, and is distributive in its operation ; 

 or, in other words, that 



V w V W <£D = V m V>D = V m+ "<£D, 



V W (</>D + ^D) = V l <t>T> + V>B. 



It is more important to observe, that it is commutative with 

 regard to x or any function of x. A slight consideration will 

 show that no kind of relation or connexion exists between y and 

 x, the former being merely a symbol denoting a change of form 

 in the functions of D. If we perform x x • $P u P on u > a func- 

 tion of x, we mean that certain operations of differentiation are 

 performed upon u, and that the result is multiplied by %x ; and, 

 since y operates upon D only, it is immaterial whether %x . <j>'D 

 be written in the form V(%# . <£D) or x#(V<£D), though the 

 latter is the more natural. We may therefore treat x and y as 

 constants to each other ; but it must be remembered that v ana< 

 D stand in the same relation to each other as D and x, and are 

 therefore not commutative. Bearing these considerations in mind, 

 it will be seen that the leading theorem above stated may be 

 placed under the condensed symbolical form, 



<KD){^a? • u}=6^{fx. 0(D)m) ; 



and the circumstance that the series is terminable whenever 

 either <f> or yfr contains positive powers only, constitutes no objec- 

 tion to this mode of writing the theorem. 



The advantage of this kind of notation in suggesting theorems 

 displays itself immediately ; for the preceding theorem at once 

 leads to the following (which I had previously deduced from 

 distinct considerations in the paper above alluded to), 

 yfrx . 0(D)w = €- D V<£(D){<ftf . u} 



= <^D{^.w}-^D{^.w}+|</> f/ D{^.M}-... 



Returning to the original theorem, and remembering that x 

 and v are commutative, we have 



(f>D{^rx . u} =e D v(<^.0(D)w) 

 =<i/r(* + V).</>(D)w 



Looking to the right-hand side of this equation, it will be 

 observed that we have extricated the function -tyx from all the ope- 

 rations ; so that the differential operations are to be performed 

 upon the function u, which thus becomes the sole subject of 

 operation ; thus tyx is no longer a part of the subject of opera- 



