This post shows how to integrate style-check.rb with flycheck in the Emacs text editor.
style-check.rb checks:
When the style-check.rb flychecker checker is functional, an example Flycheck Errors buffer is:

Download from https://www.cs.umd.edu/~nspring/software/style-check-readme.html.
To test style-check.rb, create a test.tex file containing:
\documentclass{article}
\begin{document}
\section{Heading}
The way in which.
\end{document}
Then test style-check.rb (use the correct filepath):
/path/to/style-check.rb test.tex
It should output:
test.tex:4:1: The way in which. (The way in which)
Put the following in your ~/.emacs, ~/.emacs.el, or
~/.emacs.d/init.el (or whatever) file.
Edit the path to style-check.rb.
(require 'flycheck)
(flycheck-define-checker style-check
"A linter for style-check.rb"
:command ("/path/to/style-check.rb"
source-inplace)
:error-patterns
((warning line-start (file-name) ":" line ":" column ": "
(message (one-or-more not-newline)
(zero-or-more "\n" (any " ") (one-or-more not-newline)))
line-end))
:modes latex-mode
)
(add-to-list 'flycheck-checkers 'style-check)
Enable flycheck for latex-mode:
(add-hook 'latex-mode-hook 'flycheck-mode)
style-check.rb will be executed against the latex content when you
open a .tex file. Each time you save changes, style-check.rb will
be executed.
Display the textlint warnings for file.txt in a Flycheck errors
buffer with: C-c ! l .
Debugging If flycheck errors buffer shows no warnings, debug
style-check.rb by running C-c ! C-c which will run
style-check.rb in a new buffer, showing the raw output from
style-check.rb.
Flycheck can execute multiple checkers on a single file. For example,
using the textlint flycheck check in conjunction with this
style-check.rb checker. To do this in your Emacs init file (e.g.
~/.init.el), define the textlint checker then modify the
style-check.rb checker to include:
:next-checkers ((warning . textlint))
Flycheck will now list warnings from both checkers.
A potential drawback of chaining checkers is multiple warnings about the same error from different checkers, e.g. spelling, phrasing or capitalisation.
Acknowledgement
Thanks to Blair Archibald for feedback on a draft of this post.