Solid Cylinders of Elliptic Section. 75 



The three stresses which do not everywhere vanish are 

 given by the formulae : — 



t r7(3a 4 + 2a 2 6 2 + 3//)> 2 p^ 



+ (F-- 2 ) I 3-2U^ + a 2 ^ + 2^)(l+77) 2 + 2^(a 4 -^ 4 )} J .... (17) 



^(3a 4 + 2a 2 6 2 + 3Z>0V^ 



+ (ip- z *)^ 2 {(2a i + a 2 b 2 + V)(l+ V ) 2 -2 v 2 (a i -b i )},. . . . (18) 



^(3^ + 2a 2 6 2 H-3?/)>V=-^(^ 4 + ^-^a 2 ^) (19) 



§ 4. "When 77 = the surface conditions (7) and (8) are exactly 

 satisfied, and the solution is thus in this case complete. For 

 other values of rj the solution is only approximate, but accord- 

 ing to the theory of statically equivalent load-systems, it can 

 differ appreciably from the complete solution only at points 

 whose distance from the rim does not exceed a few multiples 

 of I. Elsewhere our solution would, according to this theory, 

 seem to be correct so far even as terms of order I 2 in the 

 strains and stresses. In a thin disk, however, such terms are 

 very small and in practical calculations may be neglected. I 

 shnll thus frequently omit them, speaking of the solution 

 when they are neglected as the first approximation. This 

 first approximation is of course identical with that supplied 

 by my previous solution, as the expressions for the strains and 

 stresses in the two solutions differ only by terms independent 

 of x, y, or z and of the order I 2 . It should also be noticed 

 that the mean value of every strain or stress in the new 

 solution taken throughout the thickness of the disk is the 

 same for any given value of x, y as if terms of order I 2 and 

 Z 8 in the displacements were non-existent. In other words, 

 the strains and stresses supplied by the first approximation 

 are for every value of x and y the mean values of those sup- 

 plied by the complete solution. 



§ 5. It is most convenient to consider in the first place the 

 stress system. From (12) we see that one of the principal 

 stresses is even-where zero and has for its direction the 

 parallel to the axis of rotation. The principal stresses parallel 

 to the faces vary in direction from point to point. They are 

 parallel to the axes of the ellipse only when xy = 0, i.e., at 

 points in these axes themselves. We shall confine our atten- 

 tion to the stresses given by the first approximation. Putting 

 for shortness 



{a 4 +a 2 6 2 (l + iy)+6 4 }-s- (3a 4 +2a 2 6 2 + 36 4 ) = K, . (20) 



