Here's my proposal. To me, this feels VERY different. This feels like a team, operating together. It feels like you trust that the engineer wants to do his job and do it well. It feels like you are not imposing the date on him, but that the date is a constraint you both need to work under.
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Dear N. Gineer,
Failure report #1234 is ready for your investigation. Would you please return the attached Investigation Report to me when you've finished?
We are supposed complete this investigation by 13/12/07 so we can correct faults in a timely way and thus keep costs down. If I may assist you to avoid delays or in any other way please let me know and I will do everything in my power to help.
Thank you,
Chris
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The part about "in a timely way and keep costs down" was my attempt at reminding the person of the reason for the work to be done. Whatever the reason it *does* need to be done, perhaps you can mention or hint at it. People don't like to do things that "must be done" but they do like to do things that benefit them. In a small way, if in your email you can remind the engineer of why he should be excited to do the work (my guess was, lower costs, increase profit, increase salary) then he will be more receptive to it. Even if the reward is not financial (and focusing too much on the reward sounds like you don't trust him to do excellent work without an incentive), a simple mission of "making this company's product the best possible" could be another alternative.
My premise is that most people are motivated and want do do excellent jobs, and that what most people need are not more motivation but removal of various demotivators.
When someone gets stuck in the dreary have-to-be-done part and loses the big picture, that is demotivating. "What am I doing this for? Who cares? If I'm not trusted I may as well not act trustworthily."
Perhaps this article could get you started on some of the ideas I am talking about:
A Manager's Guide to Supporting Organizational Change: 10 Lessons Learned.