May 16, 2018 Wednesday
Bedtime Story
Note E of Ada Lovelace - Part 7
Tonight we shall resume with the Note E of
Ada Lovelace from where we had left last night.
Ada Lovelace is now describing how the
Engine would carry out its operations if the law for the general term is known.
So for the general term which is the
coefficient of (n+1)th term:
“The following would be the successive sets
of operations for computing the coefficients of n + 2 terms:-
(x, x,
,
+), (x, x, x,
, +, +), n(x, +, x,
,+)
Or we might represent them as follows,
according to the numerical order of the operations:-
(1, 2…4), (5, 6…10), n(11, 12…15)
The brackets, it should be understood,
point out the relation in which the operations may be grouped, while the comma
marks succession.
The symbol + might be used for this latter
purpose, but this would be liable to produce confusion, as + is also
necessarily used to represent one class of the actual operations which are the subject
of that succession.
In accordance with this meaning attached to
the comma, care must be taken when any one group of operations recurs more than
once, as is represented above by n(11…15), not to insert a comma after the
number or letter prefixed to that group.
n(11…15) would stand for an operation n,
followed by the group of operations (11…15); instead of denoting the number of
groups which are to follow each other.
Wherever a general term exists, there will
be a recurring group of operations, as in the above example.
Both for brevity and distinctness, a
recurring group is called a cycle.
A cycle of operations, then must be
understood to signify any set of operations which is repeated more than once.
It is equally a cycle, whether it be
repeated twice only, or an indefinite number of times; for it is the fact of
repetition occurring at all that constitutes it such.
In many cases of analysis there is a
recurring group of one or more cycles; that is, a cycle of cycle, or a cycle of
cycles.
For instance: suppose we wish to divide a
series by a series,
(1) 
or (a+bx+cx2+…)/(a’+b’x+c’x2+…)
or (a+bx+cx2+…)/(a’+b’x+c’x2+…)
it being required that the result shall be
developed, like the dividend and the divisor, in successive powers of x.
A little consideration of (1), and the
steps through which algebraical division is effected, will show that (if the
denominator be supposed to consist of p terms) the first partial quotient will
be completed by the following operations:-
(2)
{(),p(x, -)} or {(1),p2, 3)},
that the second partial quotient will be
completed by an exactly similar set of operations, which acts on the remainder
obtained by the first set, instead of on the original dividend”
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is Professor
Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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