As a verb, alternate means "to change back and forth," "to occur in successive turns." It is pronounced with primary accent on the first syllable and is usually followed by with: "Sunny and rainy days alternate with each other at this season." As an adjective, alternate is also pronounced with accent on the first syllable but is not followed by with: "He introduced an alternate proposal." Alternately, an adverb, carries the same general meaning as alternate: "The hiker alternately walked and jogged." Alternative refers to a situation involving a choice: "You have the alternative of speaking or of keeping quiet." As both noun and adjective, alternative is pronounced with primary accent on the second syllable. The distinction between alternately and alternatively is that the former suggests a meaning of "one after the other" and the latter carries a meaning of "one or the other." Neither alternative nor alternatively is restricted to a choice between only two: "The alternatives are stagnation, cold war, peace, or compromise." To keep these distinctions in mind, remember that alternate and alternately have a basic meaning of "by turns" and alternative and alternatively pertain to some kind of choice.
Dictionary of problem words and expressions. Harry Shaw. 1975.