MR. MURPHY ON THE THEORY OP ANALYTICAL OPERATIONS. 
203 
0 y + ~by) — P (ay)+~ 
d<p(ay) , 
dy ' 1 .Q 
cP (ay) 
a 2 dy 2 
+, &c. 
which being 1 also deducible from Taylor’s theorem gives the required verification. 
28. Let v 1 v 2 v 3 . . . . v n represent functions of x as multipliers, or other operations 
fixed relative to d x , it is required to find the polynomial arranged according to the 
powers (usually called orders) of d x , which shall be equivalent to the compound ope- 
ration represented by v x d x v 2 d x v 3 d x v n d x . 
It is easy to see that this polynomial cannot contain the powers of the differential 
(dv\ 2 
coefficients of any of the multipliers, that is such as(^^ ) , &c. ; moreover, if we sup- 
d n v 
pose «q = s* 1 *, then = oq” . s * lX ; thus the order of the differential coefficients of 
tq in the required result will be the same as the power of the multiplier oq" when we 
substitute s* 1 1 for «q ; and the same method will apply to discover how v 2 v 3 , Sec. are 
involved ; and when cq a 2 , &c. do not enter as multipliers we thence know that 
v l v 2 Sec. are themselves multipliers, and not their differential coefficients. This being 
considered, the question is reduced to find the value of the compound operation 
s a ‘ x d x ^ x d x n* 3 * d x z anX d x arranged according to the decreasing indices of d x , and 
tf™ v dP v 
then where we have a,™ if we put-^s 1 , and when we have a 2 p put -j^r, Sec., we 
shall obtain more readily the result of the question proposed. 
Now by the last section 
i* lX d x =(d x + a 
therefore 
£*■' d. (<*, + «,) . «<-+**>■ d, = (d, + «,) (<*. + «, + «*) £<-+*>>*. 
Similarly 
f 1 * d x f** d x f** d x = (d x + oq) ( d x + oq + “2) (^* + a i + a 2 + “3) g ( “ 1 + ai2+ “ 3)a; ; 
and generally 
^■d^’d x ^d x ..^‘d=(d I + a l ){d I + ai + u i )...{d x + a l + a 2 +...+ a y-^+----^. 
If therefore we expand according to the decreasing indices of d x , the compound 
operation, 
(d x + «i) {d x + oq + a 2 ) . . . (d x + oq + a 2 + • • • + 05 J . 2 “'* . s* 2 * . s“ 3r . . . &“ nX , 
we shall then have only to put v 1 for g* 1 *, v 2 for g® 2 *, Sec. 
^ for a, s® 1 ®, ^ for a 2 *, &c. 
dx 1 J d x 1 ’ 
To effect the composition above indicated, let us seek the product (arranged ac- 
cording to the powers of x) of 
(x + oq) (x + oq + « 2 ) (x + oq + « 2 + a 3) * * • (x + a l + a 2 + a 3 + • • • + «„). 
MDCCCXXXVII. 2 E 
