Technical Field [0001] The present invention relates to a ball bearing and methods of production of a ball bearing keeping cracking from occurring and in turn realizing an extended lifespan. 10 Background Art [0002] The various machinery used in the ferrous metal, papermaking, wind power generation, mining, and other various fields, in automobiles, and in rolling stock all use various types of ball bearings. These ball 15 bearings are used under harsh conditions under high loads, so cracks are liable to locally form. [0003] Ball bearings are standardized in, for example, JIS-B1518. In this standard, in the case of a radial ball bearing, it is recommended that a cross-sectional profile 20 line of a race at an inner race side (outer race side) (component of ball bearing defining and forming path over which rolling balls move) have a radius of curvature of a groove of 52% or less (53% or less) of a diameter of the rolling balls. On the other hand, in the case of a thrust 25 ball bearing, it is recommended that both the crosssectional profile lines of the upper race side and lower race side of races have a radius of curvature of the groove of 54% or less of the diameter of the rolling balls. However, in the ball bearings according to this 30 standard, the cross-sectional profile line of the races are arcs-having a single radius of curvature, so the contact area of the rolling balls and races at the time of a load cannot be sufficiently secured. For this reason, in particular at the time of a high load, a high 35 pressure acts on the parts of the races contacting the rolling balls and those parts are liable to crack. [0004] Therefore, in recent years, ball bearings - 2 - reducing the pressure acting on the races and suppressing cracking (Japanese Patent Publication No. 2009-174691A: PLT 1) and rolling machine elements strikingly improving the load capacity and suppressing cracking (Japanese 5 Patent No. 3608163: PLT 2) have been disclosed. [0005] PLT 1 discloses a race having a cross-sectional profile line comprised of an arc part of the same radius as the radius of the rolling balls and tangential parts formed as extensions of the arc part. According to PLT 1, 10 by selecting the circumferential length of the arc part suitably for the load, it is considered possible to suppress differential slip and sufficiently secure the contact area of the rolling balls and arc part to suppress the surface pressure applied to the race. 15 [0006] Further, PLT 2 discloses a race having a crosssectional profile line comprised of a composite arc made of a plurality of arcs with different radii of curvature smoothly connected, in which plurality or arcs, the radius of curvature at the center in the width direction 20 is relatively small and the radii of curvature at the two sides in the width direction are relatively large. According to PLT 2, by adopting this configuration, it is possible to reduce the pressure on the race. It is considered that this reduction of pressure can 25 advantageously act against the rolling fatigue of the surface of the race and against overriding of the shoulders and can reduce differential slip. [0007] Note that, in addition to PLTs 1 and 2, the technique of making the shape of the path of the rolling 30 balls a secondary curve etc. (Japanese Patent Publication No. 413'"76088: PI,T 3) and the technique of changing the radius of curvature of the curved path of the rolling balls from the groove bottom to the shoulder parts (Japanese Patent Publication No. 53-139047A: PLT 4) are 35 disclosed. Summary of Invention Technical Problem - 3 - [0008] However, when, as shown in PLT 1 (PLT 2), the cross-sectional profile line of a race which the rolling elements contact is comprised of an arc and tangents (plurality of arcs), at the time of a high load, pressure 5 excessively acts at the boundaries of the arc and tangents (boundaries of arcs with each other) and in turn cracking is liable to occur and the ball bearing is liable to be unable to prolonged in lifespan. Further, when the shape of the path of the rolling balls is 10 abstractly specified like in PLT 3 (PLT 4), it is unclear if it is possible to reliably keep down the pressure applied to the race. [0009] The present invention was made in consideration of this situation and has as its object the provision of 15 a ball bearing and methods of production of a ball bearing keeping the pressure applied to a race from becoming locally higher at the time of high load to keep cracking from occurring and in turn realize an extended lifespan. 20 Solution to Problem [0010] To solve the above problem, the present inventors in particular studied a ball bearing in which the path over which rolling elements move is defined and formed to prevent the surface pressure of the race 25 contacting the rolling elements from becoming locally higher. As a result, they obtained the discovery that if the cross-sectional profile line of the part of the race contacting the rolling elements is not comprised of a plurality of functions such as curves and curves or 30 curves and straight lines, that is, that if profile line 'i's••comprised of a specific single function,·· the suL·face pressure will no longer become locally high at the rolling elements and, as a result, cracking of the ball bearing is suppressed and in turn the ball bearing can be 35 prolonged in lifespan. [0011] Further, the present inventors also studied methods of production of a ball bearing. As a result, - 4 - they obtained the discovery that a ball bearing can be obtained by cold forging or by machining, in particular in the case of cold forging, preferably by using a specific arc shaped die. 5 [0012] Based on the above discovery, the present inventors completed the invention. Its gist is as follows: [0013] [1] A ball bearing comprising a pair of races and at least one rolling element movably clamped between 10 the pair of races, characterized in that a cross-sectional profile line of a part of each of the pair of races contacting the rolling element takes a minimum value of curvature radius at a position sticking out the most in a first direction over 15 which the pair of races face, the cross-sectional profile line becomes larger in radius of curvature the further from that position in a second direction vertical to the first direction in the crosssection, 20 that cross-sectional profile line is comprised of a single function, and when a midpoint of the profile line in the second direction is the origin, an axis extending in the second direction is the X-axis and an axis extending in the 25 first direction is the Y-axis, and a radius of the rolling element is R, the cross-sectional profile line satisfies equation (1) X'/ {2R (1+0. 05) } r ;;;> (R X d'- 307/0. 550) !/!. 28X (1 +0. 05) • • • ( 2) X2 / {2R (1+0. 05) ) pressure due to the distance from the center of groove width becomes smaller, in particular, the surface pressure becomes still smaller if the distance from the center of the groove width is 35 less than about 1 mm. [0078] FIG. 11 is a graph showing the relationship between the maximum surface pressure and the load per - 26 - ball in the different cases of the two types of FEM models shown in FIGS. 7A and 7B. As clear from FIG. 11, it was learned that when the cross-sectional profile line of the race of a ball bearing is a secondary curve, 5 compared with when that cross-sectional profile line is a single arc, it is effective to lower the surface pressure. This is important in suppressing cracking. Therefore, based on this result as well, it can be said that with a secondary curve groove, compared with a 10 single arc groove, the maximum surface pressure is lower. [0079] From the above results, according to the type of shape of ball bearing prescribed in the present application, the action of suppressing the maximum surface pressure is verified. 15 [0080] Further, in the ball bearing prescribed in the present application, the cross-sectional profile line of the race is comprised of a specific single function (secondary curve), so the cross-sectional profile line has no point not smoothly extending, so among the points 20 on the cross-sectional profile line, there are no points with surface pressures remarkably higher than other points. Therefore, according to the ball bearing prescribed in the present application, naturally there is no location at which the surface pressure is excessively 25 applied (action preventing locations where surface pressure is excessively applied). [0081] Therefore, according to the ball bearing prescribed in the present application, it can be said that the action suppressing the maximum surface pressure 30 and the action preventing locations where surface pressure is excessively applied,combine whereby at the time of load of the rolling balls etc. on a race, cracking at the race can be suppressed at a high level and in turn the lifespan of the ball bearing can be 35 prolonged. [0082] Hethods of Production of Ball Bearing Next, examples relating to the methods of production of a - 27 - ball bearing will be explained. [0083] Example Relating to Relationship of Groove Depth and Radius of Curvature of Groove Bottom From a ~90 SUJ2 rod of the composition shown in Table 1, 5 two types of test pieces for cold forging use (corresponding to race material of ball bearing) obtained after the heat treatment processes shown in FIG. 12 (units: mass%) were prepared. 10 [0084] Table 1 c Si Mn p s Cu Ni Cr Al 0 0.99 0.24 0.37 0.009 0.001 0.01 0.03 1. 42 0.012 0.0008 [0085] That is, as shown in FIG. 12, a steel material of the composition shown in Table 1 was extended by forging at 1200°C, 1 heat, from ~90 mm to ~60 mm, cut to 60~ mmx300 mm, spheroidally annealed (SA), then roughly 15 worked (outside diameter 52.5 mmxinside diameter 27.0 mmxlength 6.0 mm). Next, the roughly worked material was used as is as the finished SA material (outside diameter 52.5 mmxinside diameter 27.2 mmxlength 5.5 mm) and was quenched and tempered (QT) to obtain a finished QT 20 material (outside diameter 52.0 mmxinside diameter 27.2 mmxlength 5.5 mm). [0086] Next, the finished SA material obtained in the above-mentioned way (below, sometimes called the "SA material") and finished QT material (below, sometimes 25 called the "QT material") were cold forged. For that cold forging, a general cold forging machine (cold forging test machine with load capacity of 6000 kN) was used. Further, top dies' ·of carbidci (material: RF06) having three types of arc cross-sections of radii of curvature 30 of 5.1 mm, 4.0 mm, and 3.0 mm and ring-shaped projecting parts were used and flat plate bottom dies were used. Further, the amounts of indentation were changed to form rolling grooves in the race materials (SA materials and QT materials) to obtain races. - 28 - [0087] After forming the rolling grooves in the race materials, the shapes etc. of the rolling grooves were measured. The shape measurement generally can be performed by an optical type, laser type, or contact 5 probe type shape measuring device under conditions of a measurement width of 8 mm or more, a height resolution of 1. J.1ffi or more, and a horizontal resolution of 5 J.1ffi or more. In this measurement, a Keyence shape measuring device (VK-X150) was used to measure the shapes of the 10 rolling grooves and depths of the rolling grooves. The results are shown in Table 2. - 29 - [0088] Table 2 Sample Radius of Material Depth of Radius of no. curvature of of race groove curvature projecting (rum) of groove part of top die bottom (rum) 1 5.1 SA material 0.10 9.30 2 5.1 SA material 0.29 5.98 3 5.1 SA material 0.58 5.10 4 5.1 SA material 0.86 4.94 5 5.1 QT material 0.02 14.94 6 5.1 QT material 0.12 8.04 7 5.1 QT material 0.19 7.34 8 4.0 QT material 0.02 10.77 9 4. 0 QT material 0.14 5.48 10 4.0 QT material 0.21 5.12 11 3.0 QT material 0.02 7.77 12 3.0 QT material 0.16 3.86 13 3.0 QT material 0.25 3.69 [0089] The races of Sample Nos. 1 to 13 shown in Table 2 were investigated for the shapes of the grooves when 5 viewed by a cross-section. It was learned that each could be approximated by a secondary function. As typical examples, the cross-sectional shapes of the grooves of the races of Sample No. 4 and Sample No. 13 are shown in FIGS. 13A and 13B. 10 [0090] Furthermore, FIG. 14 is a graph showing the relationship between the radius of curvature of the bottom of the groove and the groove depth. Note that, the notations outside the box of FIG. 14 (for example, SA5.1) show the grades of the race materials and the curvatures 15 of the top die (rum) . [0091] From the results of FIG. 14, even if the material of the races and in turn the deformation resistance,differ, it is proved that the radius of' curvature R' of the bottom of the groove can be expressed 20 by equation (19) using the radius of curvature "r'' of the projecting part of the top die and groove depth ''d'': • • • ( 1 9) - 30 - [0092] Due to the above, it was proved that regardless of the material of the race, if pushing a top die with an arc shaped cross section into the race material on a flat plate, the cross-sectional shape of the recessed part 5 exhibits a profile shape of a secondary curve in each case and the radius of curvature of the groove bottom and the groove depth can be expressed by a single function. [0093] Example Relating to Effect in Case of Forming Recessed Part Race in Two Stages 10 Next, the relationship between the groove depth and the amount of indentation in the case in the "Example Relating to Relationship of Groove Depth and Radius of Curvature of Groove Bottom" wherB the radius of curvature "r" of the projecting part of the top die is made 5.1 mm 15 and cold forging a QT material is shown in FIG. 15. According to FIG. 15, it is judgBd that the material breaks when the amount of indentation is 2.3 mm. [0094] FIG. 16 is a view showing the results at the time of FEM stress analysis of a race shown in FIG. 15. 20 From the results of analysis shown in FIG. 16, an over 2500 MPa tensile stress is generated at the contact ends of the race and projecting part of the top die. It is believed that the race broke due to this stress. From the results of FIG. 15, if making the radius of curvature of 25 the projecting part of the top die smaller, the generated tensile stress becomes smaller, but for safety's sake, the maximum groove depth formed in the race material comprised of a QT material is preferably 0.2 mm or less. [0095] For this reason, for example, when desiring to 30 make the groove depth of the recessed part a final 0.6 mm,· ·:i:t ·is pref·crable to first cold forge a race matec.•al not treated by QT to form a recessed part of a groove depth of a depth of 0.4 mm or so, then treat ic by QT, then again cold forge it to further form a recessed part 35 of a groove depth of 0.2 mm or so. [0096] Due to the above, among the methods of production of a ball bearing of the present invention, - 31 - the method of production of a race by multiple stages is advantageous on the point that by forming in advance a recessed part of a certain extent of groove depth in a relatively soft material, then treating it by QT, then 5 cold forging it, even if a hard race, it is possible to precisely form a race having a desired cross-sectional profile line and in turn possible to prolong the lifespan of the ball bearing. Reference Signs List 10 [0097] 10. radial ball bearing 12. inner race 14. outer race 14a. cross-sectional profile line of path 16, 26, 32. rolling balls 15 20. thrust ball bearing 22. upper race 24. lower race 34, 36. races D. groove depth 20 Dl, D3. first direction D2, D4. second direction L. distance from groove bottom Pl, P2. points W. 1/2 groove width CLAIMS Claim 1. A ball bearing comprising a pair of races and at least one rolling element movably clamped between said pair of races, characterized in that; a cross-sectional profile line of a part of each of said pair of races contacting said rolling element takes a minimum value of curvature radius at a position sticking out the most in a first direction over which said pair of races face, said cross-sectional profile line becomes larger in radius of curvature the further from said position in a second direction vertical to said first direction in said cross-section, said cross-sectional profile line is comprised 15 of a single function, and when a midpoint of said profile line in the second direction is the origin, an axis extending in said second direction is the X-axis and an axis extending in said first direction is the Y-axis, and a radius of said 20 rolling element is R, said cross-sectional profile line satisfies equation ( 1) : X'/ {2R (1+0. 05~ )