So long as the bearings are new and properly made, each ball touches and rolls along what may be considered a mathematical line, and there is, in fact, no friction worthy of consideration. Nevertheless there is some, and in time a small groove is worn, or rolled, into the bearing, which groove just fits the ball. The friction is greater now than before, and increases with the deepening groove until, finally, when the depth of the groove equals the radius of the ball, the friction reaches its maximum and would be at that time nearly equal to one-fourth of the amount of friction engendered if the ball actually slid in the groove. The ball would then roll on lines along the groove through points _c_, _c_ thirty-eight and one-fourth degrees around from _E_ towards _D_, as shown in the annexed diagram. (Fig. 1.) [Illustration: Fig. 1. Rolling Lines, ball-bearing.] The reader can form a tolerably clear idea of the amount of friction caused by the ball sliding without rolling; let this then be the unit. Also let the radius of the ball be the unit depth of groove. The following table gives roughly in these units the frictions for the groove depths expressed in tenths. +---------------+---+---+---+---+---+---+---+---+---+---+---+ |+Groove Depths+| 0 |.1 |.2 |.3 |.4 |.5 |.6 |.7 |.8 |.9 |1.0| |+Frictions+ | 0 |.01|.02|.03|.05|.07|.09|.12|.15|.18|.21| +---------------+---+---+---+---+---+---+---+---+---+---+---+ To what is this friction due? Look at this diagram (Fig. 2), representing a transverse section of the groove and ball. [Illustration: Fig. 2. Transverse section groove and ball.]