180 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
2. The operations written as above are monomial , consisting of only one term ; and 
polynomial operations give the sums or the differences of the results of the respective 
monomials of which they are formed, according as these monomials are affected by 
the signs + or — . 
Thus if 1 as an operation be understood as the multiplying of the subject by unity 
which leaves it unaltered, and the symbols p, A have the same signification as in 
art. 1, then 
[u] (P — 1) = O] A 
O] (A + 1) = O] P, 
where the subject u is any quantity whatever. 
When general relations, such as these, between different symbols exist independ- 
ently of the particular value of the subject, we may abstract the consideration of 
the latter, and the sign = between symbols of operation being understood to indicate 
that they are universally equivalent, the symbols used in art. 1 would have the fol- 
lowing relations independent of the subject. 
A = 4- — 1, '4/ = A-J-l, ^ — A = 1, 
when A is put = 0. 
3. A compound operation consists of a series of simple defined operations, mono- 
mial or polynomial, the subject of each individual in this series after the first being 
the result of all the preceding operations. 
Thus [x 2 ] a p A d x denotes that first x 2 must be multiplied by a, which gives a x 2 ; 
then the operation p, which denotes the putting x -J- A for x, gives a (x + A ) 2 ; next 
the operation A will give a (x + 2 h ) 2 — a (x -j- h) 2 , or a (2 A x + 3 A 2 ) ; and lastly, 
the symbol of taking the differential coefficient relative to x gives 2 ha: the final or 
complete result is therefore 
[x 2 ] a p A d x — 2 A a. 
When all the symbols in a compound operation are exactly the same, then for 
abridgment the whole operation is represented by writing an index to the right of the 
symbol for the simple operation over it, this index expressing the number of times 
the simple operation is repeated. Thus 
[ x 2 ] p 3 — [x 2 ] p p p = [(# + A) 2 ] p p/ = [(# -f- 2 A) 2 ] p = (x -f- 3 A) 2 . 
But when the simple operations are different, they must be written consecutively 
in the order in which they are to be performed, unless that order of arrangement by 
the mutual relations of the operations should be indifferent. Thus 
[x n ]xp= [> n + 1 ] p = (x+ A) n + 1 
jV'] p x = [(a? -f- A)”] x = x {x + A) 71 . 
