A surgical drilling instrument with a Curved Burr Attachment having an adapter removably coupled to a drill motor and a burr attachment coupled to said adapter. The burr attachment includes a tube with a wire shaft supported therein and a coil wire surrounding said wire shaft and having each of the coils touch the outer periphery surface of the wire shaft and the inner peripheral surface of said tube to enhance cooling of said outer periphery of said tube from the heat generated by said wire shaft and attenuate vibrations while allowing said burr attachment to be bent without kinking.