Solid Cylinders of Elliptic Section. 89 



that material. The velocity is chosen merely as a convenient 

 number, and there is no intention to imply that it falls in each 

 or in any case within the limits wherein the mathematical 

 theory is applicable. If any other velocity exist in a disk of 

 the material, we have only to remember that all the quantities 

 in the table vary as (cod) 2 . For instance, we have only to 

 multiply the entries in Table V. by "16 to get the results for 

 a velocity of 400 feet per second. The reader may also easily 

 deduce from the table results for any other material in which 

 rj = "25. He has only to notice that all the five quantities 

 tabulated vary directly as p, and that all except the maximum 

 stress-difference — which depends only on (coa) 2 p — vary in- 

 versely as E. 



§ 23. The radial displacements and strains — L e. those di- 

 rected along the radii vectors perpendicular to the axis of 

 rotation — appear the most interesting of those in planes 

 parallel to the faces. According to the first approximation 

 the radial displacement u r and strain s r at a point (#, y), or 

 (r, 6), are independent of z. They maybe deduced from (14) 

 and (15) by means of the relations : 



ru r =xa+y(3, ~\ 



7 S *- X dx + XlJ \dy + d X r y dyJ 



Confining ourselves in the meantime to the first approxima- 

 tion, let us for shortness write (14) and (15) as 



B = B 1 y-iB 3 y*-B 3 iyrf /' ^ 



and put 



3(A 3 ' + B 3 ') = C 3 (52) 



Then we have 



ujr = A x cos 2 + Bi sin 2 -£r 2 (A 3 cos 4 <9 + 



C 3 sm 2 0cos 2 + B 3 sin 4 <9), (53) 



s r =Ai cos 20 + B^in 2 <9-?- 2 (A 3 cos 4 + 



C 3 sin 2 0cos 2 + B 3 sm 4 0) (54) 



Thus a point is displaced away from or towards the axis of 

 rotation according as it lies inside or outside of the curve 



r 2 = ^ 1 2 = 3(A 1 cos 2 ^ + B 1 sin 2 6')^(A 3 cos 4 (9 + 



C 3 sin 2 0cos 2 0-HB 3 sin 4 0), (55) 



