Concept:
The Cotton Count (also called the English Cotton Count or Ne) is an indirect system of expressing yarn fineness.
In an indirect count system,
\[
\boxed{\text{Finer the yarn } \Rightarrow \text{Higher the count}}
\]
The English Cotton Count is defined as:
\[
\boxed{
N_e=\frac{\text{Number of hanks}}{\text{Weight in pounds}}
}
\]
where,
• One hank of cotton yarn \(=840\) yards.
• The weight is measured in pounds.
Since the weight is kept fixed (1 pound), the cotton count is known as a constant weight system. The length of yarn varies according to the fineness of the yarn.
Step 1: Understand the meaning of 40's Cotton Count.
A yarn having
\[
40's \; \text{Cotton Count}
\]
means that
\[
40
\]
hanks, each of length
\[
840 \text{ yards},
\]
together weigh
\[
1 \text{ pound}.
\]
Thus,
\[
\text{Total Length}
=
40\times840
=
33600 \text{ yards}.
\]
Therefore,
\[
\boxed{
40 \text{ hanks weigh }1\text{ pound.}
}
\]
Step 2: Identify the counting system.
Since
\[
\boxed{
\text{Weight remains fixed}
}
\]
and the length changes with the count, the English Cotton Count belongs to the
\[
\boxed{\text{Constant Weight System}.}
\]
This is why finer yarns have higher count values.
Step 3: Examine the given options.
Option (A): Constant weight
This is correct because the English Cotton Count keeps the weight fixed at one pound.
\[
\boxed{\text{Correct}}
\]
Option (B): Constant density
Density is unrelated to the definition of cotton count.
\[
\boxed{\text{Incorrect}}
\]
Option (C): Constant length
Constant length systems are direct count systems such as Tex and Denier.
\[
\boxed{\text{Incorrect}}
\]
Option (D): Constant volume
Volume is not considered in yarn count systems.
\[
\boxed{\text{Incorrect}}
\]
Step 4: Write the final answer.
Hence,
\[
\boxed{
40's \text{ Cotton Count refers to the Constant Weight System.}
}
\]
Therefore,
\[
\boxed{\text{Option (A) is the correct answer.}}
\]