It may work that way but its not proper, switches are very inexpensive these days. There's two problems I see with that configuration.
1. DHCP scopes being the same, which can cause conflicts.
2. Not having the ability to access computers behind the second router.
Here's a solution if you really don't want to buy a switch, kind of goofy in my opinion since you would only be gaining three ports but you don't have to hook it up to the WAN port on the second router. Hook up the second router by itself first, login to the web interface and disable DHCP and the WAN port if possible, then you can connect the two routers together without having to worry about IP conflicts through the LAN ports.
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