In Summer of the Seventeenth Doll by Ray Lawler, what is the third quality that Barney says a woman must have to appreciate a man like him?
Since Pearl does ask about the third quality, but Barney does not say what it is, readers are at pains to understand what that quality is and must infer from the text and context what it might be. Considering what we learn in the fight scene between Roo and Barney, preceding the conversation between Barney and Pearl, one reasonable inference is that the unspoken third quality is deep love for him. Then, considering what we learn in a later scene between Roo and Emma, one reasonable inference is that the quality is shrewdness: being shrewd enough to see the reality of life and of true unity (marriage). Putting both inferences together, we might say that the quality is that of choosing real life over illusion, even the illusion of sixteen years of summer love. Pearl wouldn't like this answer because she has her own illusions that she clings to.
Examining these scenes in some depth, the fight between Roo and Barney, ending Act II and preceding Pearl's Act III departure, offers some insight into Barney's perception of what that third quality is (the first two qualities are given haltingly at the end of Act I). From what we learn through their fight, it is clear that Barney loves Nancy because of the rage he goes into when Roo brings Nancy into his verbal assault on Barney. Barney is humiliated, though feebly attempting to defend himself all the while that Roo grapples him and roughly restrains his arm, pushes him to the floor, drags him back to his feet, then shakes him. Even after Barney tears himself free, "'S enough, Roo," he still is not in the same rage that has overcome Roo. It is only when Roo taunts Barney with his loss of Nancy that Barney is overcome with rage and strikes back:
ROO: And Nancy--after seventeen years, you couldn't even hold Nancy.
BARNEY: You rotten--your dirty rotten--
[Angry beyond measure, BARNEY seizes the nearest heavy object to hand. ... BARNEY swings this at ROO's head.]
To Barney, Nancy is not like any other woman he has taken up with. She is one of a kind in his mind. This, of course, is painfully ironic--and the probable source of Barney's loss of virility catalogued so loudly and hatefully by Roo--since Nancy's marriage leaves Barney on his own during the seventeenth magical, youthful lay-off season. When Pearl asks Barney about the other quality, he says it is "far too late" for her to be asking, that she "wouldn't like" the answer if told, and that he'd only "met one woman in [his] life who ever had" it. Barney's implication is clear to Pearl and to us: that woman is Nancy.
BARNEY: Only met one woman in my life who ever had it, anyway.
BARNEY: Her. And she went off and got herself married months ago.
When Roo and Emma later talk about Nancy--providing background to the foursome relationship--Emma says that Nancy is shrewd, shrewd enough to "buy and sell Olive any day." Responding to Roo's question about whose fault it was that the seasons of fun were messed up and came to a "cropper," Emma says the "lay-off seasons" were "just--seasons" and "not for keeps"; the four of them had "gone as far as" they could go with their summer lay-offs (remember that in the Southern Hemisphere summer months come at the end and beginning of calendar years, roughly November through February: "New Year's Eve. A hot velvety summer night.").
This conversation suggests that the third quality is the shrewdness Nancy has: being shrewd enough to see the reality of life, she is shrewd enough to choose love of real life over love of one "lay-off season" cane cutter. Both inferences produce the one idea that the quality is being able to love deeply yet choose reality over illusion; being able to see when something "not for keeps" has gone as far as it can go.