Found the solution here:
http://stackoverflow.com/questions/22063520/laravel-slash-after-url-redirects-to-root-folder
Basically the .htaccess that laravel ships with will redirect any matching path with a trailing slash to "/path". You can get away with it using a RewriteBase and setting that rewrite rule to use that base. So now you have this in .htaccess:
RewriteRule ^(.*)/$ /$1 [L,R=301]
That needs to be changed (for the OP example) to this:
RewriteBase /2ndApp
RewriteRule ^(.*)/$ $1 [L,R=301]
This assumes that "2ndApp" is the directory that starts its life as "public" when you first install Laravel.
The RewriteBase would need to be set differently for each installation into different root directories, but does the job.