It's been several months since I opened this thread. Correction: my model is 10515, not 10415 as previously stated. The replies I received were very helpful. I downloaded the service manual from the Lawn Boy Support Center. The manual says to use 6 oz. (177cc) No. 2 lithium-based grease as a lubricant for the three speed transmission. The instructions say to coat all moving components during assembly, and then to fill the gear case with 6 oz. (177cc) No. 2 lithium-based grease. Also, to apply a light bead to the (gear box) flange before assembling. I guess that serves as a kind of seal between the gear box and the cover.
I followed the thread in reply No. 2 and found all kinds of suggestions for types of grease to use. None of them even mentioned lithium-based grease. In the meantime, I ordered a complete gear box from a used 10515 on ebay, I plan to overall it using the No. 2 lithium based grease. If it's good enough for the official Lawn Boy service manual, it's good enough for me. And besides. I already have a can of it in my garage.