Is your Mac Intel or PowerPC?
Mount the DMG, and open the folders to the 10_3_r183_90 folder.
Mount the flashplayer_10_3r183_90_mac_intel.dmg.
Right click the Flash Player installer (or Ctrl+click)
Choose Show Package Contents.
go to: Contents/Resources.
Double click the Adobe Flash Player package.
It should install from there.
A side note: Being as your Mac is Intel, you really should consider upgrading to OS 10.6 (Snow Leopard). It will enable you to run a whole lot of software that 10.5 can't, including the latest Flash Player and Safari 5, which is a better browser than Safari 4. It's only $19.99 through Apple and they ship free. By doing an upgrade with the migration assistant, you keep all your files, and only the OS upgrades. All passwords and logins stay the same.
Upgrading to OS 10.6 when I receive it on Monday. Then I will try to
install the newer Flash.
Thank you for your help.