The legal case revolves around the computer firm\'s use of the Apple name and logo in branding its iTunes online music site. The website allows iPod users to download and save thousands of songs.
Logo rights
The Apple Corps logo is a green Granny Smith apple, while the computer firm is represented by a multi-coloured striped apple with a bite taken out.
In 1991 both companies reached an agreement over the trademark that kept Apple Computer from entering into the music market and also barred Apple Corps from the computer, telecommunications and data processing industry.
Created by rock-n-roll legends The Beatles, Apple Corps is currently owned by former members Paul McCartney, Ringo Starr and the widows of band-mates John Lennon and George Harrison.
The lawyer for the music company, Geoffrey Vos, said the deal gave Apple Corps the exclusive right to use "apple" marks for the record business. He added that Apple Computer had reciprocally rights for the information technology market.
Unforeseen problem
The fifteen-year-old deal was in part set up to offset any competition that involved compact discs and tapes and did not anticipate technological developments such as iTunes.
But the deal was threatened in 2001 with the launch of Apple Computer\'s iPod. In 2003 the iTunes music download website hit the internet.
Mr Vos told the court the electronic download system perverted the constraints in the 1991 deal. He claimed Apple Computer founder Steve Jobs had said that downloading music from the Internet was the modern equivalent to buying records.
The plaintiff argued that the US firm had violated the deal by selling music online reaching a one billion downloads milestone through their iTunes website.
Apple Computer is the market leader for music downloads, with around three million songs being downloaded each day from its iTunes service.
The Beatles\' company is seeking financial damages and court orders to stop Apple Computer using the "apple" marks in connection with the iTunes website.
The hearing, expected to last a further four days, was adjourned until Thursday.