He has also urged them to vote against a resolution put forward by a group of underwriting members protesting about the pounds 2bn losses that have hit the market.
Following this week's extraordinary general meeting a postal vote is taking place over the next month on the five resolutions, supervised by the Electoral Reform Society.
In his letter to the 30,000 or so members eligible to vote, Mr Coleridge says: 'I urge you to use this opportunity to vote in the best interests of your Society. Rapid changes are taking place at Lloyd's. Your council is committed to a programme of reform which, I am certain, will serve the interests of all our members.'
Mr Coleridge also restates his endorsement of David Rowland, chairman of Sedgwick Group, the large insurance broker that places a significant amount of its business at Lloyd's, as his successor when he steps down at the end of this year.
'I am delighted, however, that the council has accepted my recommendation that David Rowland should succeed me subject to his election to the council.' He adds: 'I am confident that you will support his leadership of the Society over the next few years.'
The resolution for which Mr Coleridge is seeking support expresses confidence in the council but urges Lloyd's without further delay to implement reform proposals contained in an internal report prepared by a task force headed by Mr Rowland. The task force came up with a series of recommendations on how Lloyd's should operate over the next five to seven years.
The resolution, sponsored by the Association of Lloyd's Members, representing the interests of more than 8,000 members, called on Lloyd's to implement the task force's recommendations, including the separation of market governance from regulation and the formal recognition of the primacy of members' rights and the changes of attitude and behaviour that would result from this. The resolution also called on the working community in the market 'to make a substantial financial contribution to the problem of distressed names (the members).'
The resolution caused dismay among many members of the association who felt that a vote of confidence was inappropriate. Neil Shaw, chairman of the association, defended the decision this week. 'What we are saying is that we have no confidence in the council unless they change themselves. We framed the resolution in this way so the council could support the resolution yet commit itself to the changes,' he said.
In Canada, 70 underwriting members taking legal action against Lloyd's over their losses suffered a setback. The Court of Appeal for Ontario has reaffirmed that cases involving Lloyd's should be heard in English courts under English law. The 70 members had alleged fraud by Lloyd's and hoped to have their case heard in Ontario.