The word “equipment” is a mass, noncount noun that in normal usage can be both singular or plural depending on whether it refers to a single unit or to several units of that equipment. Being noncount, it does not need the suffix “-s” in its plural form but stays as is—“equipment.”
In English, a noun preceded by the adjective “other” must be in its plural form, as in “Other girls are joining the picnic.” In the case of “equipment,” of course, it is already plural even without “-s” as suffix.
Also, when a verb in English is in the passive-voice future tense, it takes this form in a sentence: “will be” + past perfect form of the verb. The verb “purchase” thus takes the form “will be purchased.”
So, assuming you mean to say that several units of that “equipment” will be purchased, the grammatically correct phrasing of the statement you presented would therefore be the following:
“...other equipment will be purchased after the second quarter...”
The phrasing will be the same even if the article “the” precedes that statement to indicate that only a single additional unit of that equipment instead of several will be purchased:
“...the other equipment will be purchased after the second quarter...”