The U.S constitution is the supreme law of the land, higher than any other. The U.S constitution gives the senate the right to judge the qualifications of its members. Right here in Article One, Section 5:
"Each House shall be the Judge of the Elections, Returns and Qualifications of its own Members, and a Majority of each shall constitute a Quorum to do Business; but a smaller number may adjourn from day to day, and may be authorized to compel the Attendance of absent Members, in such Manner, and under such Penalties as each House may provide."
http://www.usconstitution.net/const.html#A1Sec1As well, same article and section:
"Each House may determine the Rules of its Proceedings, punish its Members for disorderly Behavior, and, with the Concurrence of two-thirds, expel a Member."
http://www.usconstitution.net/const.html#A1Sec1Therefore, the Senate is given the authority to not only judge the qualifications of its members, and set rules of how the senate is to be run.
The Senate does, in fact, have these rules of its proceedings. Like Section 2, Rule 2:
"2. The Secretary shall keep a record of the certificates of election and certificates of appointment of Senators by entering in a wellbound book kept for that purpose the date of the election or appointment, the name of the person elected or appointed, the date of the certificate, the name of the governor and the secretary of state signing and countersigning the same, and the State from which such Senator is elected or appointed. "
http://rules.senate.gov/public/index.cfm?FuseAction=RulesOfSenate.View&Rule_id=45831f40-f4a1-4348-b332-63479dd3a55a&CFID=57419949&CFTOKEN=59867292The last sentence details the necessity of a SoS countersigning the certificate of appointment or election. White has not signed said certificate. Therefore, according to constitutionally allowed senate rules, the senate has used its constitutional right to judge the qualifications of its members, and has found Roland Burris to not meet the qualifications for membership.