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The phrase "a second description" is correct and usable in written English.
It can be used when referring to an additional or alternative description of something that has already been described once.
Example: "In the report, I provided a first description of the process, and now I will offer a second description to clarify the details."
Alternatives: "an additional description" or "a further description".
Exact(3)
Deryk James, acting for the family, offered a second description for the cause of death: "Sudden death in a man with pulmonary embolism and squamous carcinoma of the bronchus during struggle against restraint".
A second description of natural law is: the first principles or starting points of practical reasoning.
A second description of a C ∗ -algebra, from an operator-theoretic point of view, is given here.
Similar(57)
(A first description of the RCM, modelling random-access channels only was provided in [25], and [26] briefly outlines the ideas of using clusters for time-variant channel modelling).
A first description of the expression values showed two problems: censored values and values similar to background values.
In summary, here we provide a first description of sex specific gene expression differences in human and murine kidneys under baseline and disease conditions.
Here, we present a first description of a X. tropicalis cell line.
The analysis described in this paper provides a first description of marriage and partnership patterns and trends in this population.
Inclusion criteria for scoring systems were a first description of a multiparametric, semiquantiative scoring systems which comprehensibly describe an approach to evaluate morphologic lesion.
First, we aimed to provide a first description of the spatio-temporal pattern of all PRRS cases in Thailand in 2010.
Waters and co-workers provided a first description on the main events characterizing the shift in gene expression in this tissue around véraison [ 22].
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com