82 Mr. C. Chree on Rotating Mastic 



The simple relation 



_1 1 _ 1 + y 1 _ 7){l + r))co 2 p f33 v 



a l **b 1 *~a* + l> i K'- EM K J 



will subsequently be found of service. 



It is easy to prove for all permissible values of b/a and rj, 



a ± >a, bi>b. 

 Thus by (29) every point in the disk approaches the central 

 plane z=0. Those points originally in a plane section 

 parallel to the faces, which are also during rotation equi- 

 distant from the central plane, lie on one of a series of elliptic 

 cylinders whose common axis is the axis of rotation, and 

 whose cross sections are similar and similarly situated to 



tf 2 /V + */W=l (34) 



The common eccentricity e x of these sections is given by 



^ = l_(6 1 /a 1 )2 = e 2 ( 2-^ ) (1 + 3l7 ) -+-{4(l+^)-*(3+i)+*}. 



The major axis of (34) lies along the major axis of the disk, 

 but e v is easily proved less than e for all permissible values 

 of e and rj. Ellipses similar and similarly situated to (34) 

 drawn in the central plane may, for the sake of reference, be 

 termed ellipses of equal longitudinal displacement. 



§ 15. For points originally in a plane section parallel to 

 the faces z is constant, and thus by (29) such a section is, 

 according to the first approximation, deformed into a para- 

 boloid whose axis is the axis of rotation. The paraboloid is 

 elliptic for all permissible values of b/a and tj, and its con- 

 cavity is directed away from the central plane. Over the 

 faces themselves the terms in (16) neglected by the first 

 approximation vanish ; thus the paraboloidal form of the faces 

 is exact so far as our solution goes. The curvature in any 

 plane through the axis of rotation at the vertices of the para- 

 boloids into which the plane sections transform is, by (29), 

 directly proportional to z, and so is greatest in the faces of 

 the disk. Also, as b± < a 1} the curvature in any given para- 

 boloid is greatest in the plane containing the minor axis of the 

 disk, and least in that containing the major axis. At the faces 

 the measures of the curvatures supplied by (29) are the same 

 as those supplied by the more exact solution (16). Denoting 

 the principal radii of curvature at the centres of a face by 

 R x and R 2 , the former being taken in the plane xz, we have 

 by (29) 



R 1 = a 1 2 /(2M0, R 2 =6 1 2 /(2M/), .... (36) 

 where M is given by (30). Hence, employing (33), we 



