MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
197 
A remarkable property of intermediate operations is that they are also intermediate 
between any operations which are the same functions of the extremes. 
For let 
9 i ~ i z 
then performing the operation z 
9 1 z — i z 2 
put now for t z its equivalent operation 9 1 , and we have 
9 2 i = / z 2 . 
Similarly if we suppose 
9 n ~ 1 i = iz n - 1 
then 
but 
therefore 
9 n ~ l iz = iz n 
i z — 9 
9 n i = iz n 
Again, suppose the subject in the last equation to be one on which the opera- 
tion 9~ n has been performed, then that equation becomes 
/ = 9~ n i z\ 
or 
9~ n i — tz~ n . 
Again, suppose K an operation satisfying the equation 
n 
0m < = ( K. 
We have by the parts of the proof already given in this article, 
rtfl V Tl 
H ( — S l\ = l Z , 
or 
K m = z n ,K = Zm ; 
hence 
n n 
9’fii l — i zm . 
From these premised equations it follows, that if f (9) f (z) represent the aggregates 
of any similar powers of the operations 9, z , with the same coefficients, we must have 
generally 
f(9).i = #/(*). 
By this theorem, if f (9) be known, / (z) can be found, supposing that we know i the 
operation intermediate to 9, z. 
21. We shall now apply this theorem to cases where 9, t are given, and therefore ^ 
known, as above shown. 
Let one extreme 9 represent the operation of differentiating relatively to x, and the 
intermediate / that of multiplying by z ax , then we have 
d x z ax = z ax z 
2 d 2 
