Brian signed the orginal contract for the Rolling Stones (the band) with Decca in May 1963. He was the representative for the band. In July 1965 the band (The Rolling Stones) renegotiated a five year deal. Jones was still the rep/owner of the band The Rolling Stones"..But all five members signed the contract. The band "The Rolling Stones" could not change it had to be Jagger/Richards/Wyman/Watts & Jones until July 31 1970. That is why Mick Taylor was not actually an official member (the 5th Rolling Stone) on paper and was only on salary until August 1, 1970. Although when the 69 tour was over I believe Mick T. got a fifth of the profits split evenly like an official band member.
In May 1969 when Jones "left" the band he still maintained the name "The Rolling Stones" and the band the Rolling Stones would still be Jagger/Richards/Wyman/Jones & Watts. After "leaving", Brian still could if he wanted form another band with other artist and keep the name "The Rolling Stones". We have seen this years later with other groups... there would be two or three bands touring with the same name like the Shangra La's, Sam & Dave and I believe the Temptations among other artist. Not a good thing for the band "The Rolling Stones" about to embark on a tour of the US in Nov. 1969. Well this never became a problem as Brian died in July 1969 before he could form another band.
But here is the surprise. Our Rolling Stones of 1962/63 ..really didn't own the name.. The name The Rolling Stones was owned by another band up in Northen England, they actually had the name registered/trade marked. They only foud out about our Rolling Stones when all of a sudden they were getting turned down at venues saying "we don't want your kind playing at our place".. When they found out about this other "Rock & Roll" band using their named they sued the Rolling Stones (probably Easton & Oldham) as well in 64/65 (I have the article about the suit, I'll read it again). Our Rolling Stones ignored the suits over & over, knowing they really didn't have a claim to the name.. But what eventually happened was the Offical band named "The Rolling Stones" after awhile stopped suing and broke up, leaving our "The Rolling Stones" owning the name I guess by default.
There has been a history of Bands named the Rolling Stones another U.K. band had the name "The Rolling Stones" in the late 20's early 30's. (there is even a photo of them) In the U.S I found a record released by "The Rolling Stones" from the late 40's early 50's I believe it ws a country band, by the title; if I can find it I will post the title and label. There was another band from Chicago for a short time I believe around 1954/55 that called themselves The Rolling Stones" for about 8 months to a year. One of their band members was the famous Luther Allison who later played with Howlin Wolf and Freddie King among others. So in the end I guess you could have started a band in 1963 and called yourself "The Rolling Stones" and might have been able to keep the name(If you wanted to fight it in court forever
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