76 Mr. C. Chree on Rotating Elastic 



we may resolve the stress system at any point into the fol- 

 lowing simple systems : — 



(i) A uniform normal tension &) 2 /)a 2 K parallel to the major 

 axis. 



(ii) A uniform normal tension co 2 pb 2 K parallel to the minor 

 axis. 



(iii) A normal pressure — co^pr^K directed along the 

 radius r of the circle concentric with the disk which passes 

 through the point in question. 



(iv) A normal pressure directed along the tangent at the 

 point considered to the ellipse which passes through the 

 point and is similar and similarly situated to the rim of the 

 disk. If a', V denote the semi-axes of this ellipse, p' the 

 perpendicular from the centre on the tangent, this pressure is 



-»w+H M + si* +w (fy. . . (2D 



At the rim of the disk the principal stresses in the plane 

 through the point considered parallel to the faces are directed 

 along the normal and tangent to the rim respectively. The 



former stress is everywhere zero, and the latter, tt, is given by 



~ cfi + V^aW / ah Y m , 



tt ^ COp 3a^2a%^S^{^ )' ' * ' (22) 



where p is the perpendicular from the centre on the tangent 

 to the rim at the point considered. Since rj cannot exceed 

 •5 this is everywhere a tension. Its maxima are found at 

 the ends of the minor axis, its minima at the ends of the 

 major axis. 



Without entering on a discussion of the terms of order I 2 

 in (17) and (18), it is worth noticing that they indicate that 

 for a given value of x, y the traction, algebraically considered, 

 is greatest in the central plane and diminishes from thence 

 to the faces of the disk. This seems to indicate that the 

 tendency in the material to retire from the axis of rotation is 

 greatest in the central plane. 



§ 6. One of the most important points in such problems as 

 the present is the determination of the greatest speed con- 

 sistent with safety. Unfortunately, our knowledge of the 

 conditions requisite for safety is very incomplete, and it is 

 at least doubtful whether the question comes in general 

 within the scope of the mathematical theory. But admitting 

 the doubtful character of existing theories, it is desirable to 

 examine the conclusions they lead to, if only for the reason 

 that fresh light may thus be thrown on the question of their 



