90 



Mr. C. Chree on Rotating Elastic 



and the radial strain is an extension or a compression accord- 

 ing as the point lies inside or outside of the curve 



f*=£f-i*=ffc* say (56) 



The curves r=ri and r=r 2 are here termed the curves of 

 . no radial displacement and no radial strain respectively. The 

 relation r^ = 2>r^ enables most properties of one curve to be 

 immediately deduced from those of the other. 



§ 24. For disks of small eccentricity the curve (55) lies 

 completely outside the perimeter of the disk ; but as b/a is 

 diminished its radius-vector in the direction of the minor axis 

 becomes gradually reduced to zero. The values bja and 

 b 2 /a of b/a, for which the radius-vector of (55) drawn in this 

 direction reaches the values b and respectively, are derived 

 from the equations 



3B 1 -6 2 B 3 =0 ? (57) 



B 1 =0,i.e.& 8 /«=<\A; (58) 



see (51). 



These values are shown in Table VI. for a series of values 

 of 77. 



Table VI. — Values of b\ja and b 2 /a. 



n= 



0. 



•2. 



•25. 



•3. 



•5. 



bja = 







•519 



•5725 



•619 



•765 



bja = 







447 



•5 



•548 



•707 



The results are all only approximate except those for rj = §, 

 and the value of b 2 /a for 77 = '25. When b/a is greater than 

 bja, every point in the disk increases its distance from the axis 

 of rotation. When b/a lies between b-Ja and b 2 /a, the curve 

 of no radial displacement cuts the minor axis at a finite 

 distance less than b from the centre. The curve then appears 

 constricted in the diameter measured along the minor axis, 

 and this constriction rapidly increases as b/a approaches b 2 /a. 

 When b/a is less than b 2 /a the centre is a double point on the 

 curve, and its tangents there are inclined to the major axis at 

 an angle 6 given by 



flo^tan-^-A^Bi), .... (59) 



where A x and Bj have the same meanings as in (51). 



When the centre is a double point on the curve of no radial 

 displacement it is obviously by (56) also a double point on 



