"a cat without a tail"
Female Manx Cat
Public DomainFemale Manx Cat - Credit: Joachim JesuBek, Wikimedia Commons
A long-haired Manx (also known as a 'Cymric')
Creative Commons AttributionA long-haired Manx (also known as a 'Cymric') - Credit: Robertlucien, Wikimedia Commons

As becomes clear a few lines later, this is a reference to a Manx cat, a tail-less breed which originated  on the Isle of Man where it remains common.

 

 

 

 

 

At this point in the text, the Manx cat ('abrupt' and 'truncated') becomes a symbol to the narrator of something that is lacking, or different, and this leads on to a chain of complex associations, a characteristic aspect of  Virginia Woolf's fictional style. 

Little Manx cat to fluffy friend:
Creative Commons AttributionLittle Manx cat to fluffy friend: "Do you know, I don't think I've ever felt as abrupt and truncated as I do today" - Credit: Janice Waltzer, Flickr