By lifting the second finger off the 5th string you're adding an 'a' to the E chord which consists of the notes e-g#-b. Chords are usually (always?) made up of thirds, meaning there's either 3 or 4 semitones between each note. In the case of major chords like our E here, there's a major third (4 semitones) between 'e' and 'g#' and a flat third (3 semitones) between 'g#' and 'b', as is the case for all major triads (A, C, D, E, G etc.). The 'a' you've added is only a semitone higher (minor second) than 'g#' and 2 semitones lower (major second) than 'b'. To my, admittedly very limited, knowledge of music theory, there is no type of chord that is made up of those intervals (major third-minor second-major second).
I'm still very much a beginner in music theory, but this is how I understand chord construction right now. I'm sure someone will correct me, if what I outlined above is not always true.
Edit: What richard posted is of course, also true. By omitting the 5th and the 6th string from strumming, you're still only playing the notes e-g#-b, just on two less strings.
