216 



Mr. C. Chree. 



[May 17, 



posing no bodily forces to act. The solution involves arbitrary con- 

 stants, and consists of a series of parts, each composed of a series of 

 terms involving homogeneous products of the variables, such as 

 x l y m z n ~ l ~ m , where I, m, n are integers, and n is greater than 3. The 

 case n = 7 is worked out numerically as an illustration. fc The terms 

 involving powers of the variables, the sum of whose indices is less 

 than 4, are then obtained by a more elementary process, and these 

 alone are required in the applications which follow. These terms 

 arrange themselves in groups associated with certain constants in the 

 expression found for the dilatation. 



The first application of the solution is to " Saint-Venant's problem " 

 for a beam of elliptical cross-section. The problem is worked out 

 without introducing any assumptions, and a solution obtained, which 

 is thus directly proved to be the only solution possible if powers of 

 the variables above the third be neglected. Certain groups of 

 associated constants vanish completely, and the remaining arbitrary 

 constants express themselves very simply in terms of the terminal 

 forces, all the constants of one group depending on one only of the 

 components of the system of forces. 



Part III consists of an application of the second portion of the 

 solution of Part II to the case of a spheroid, oblate or prolate, and 

 of any eccentricity, rotating with uniform angular velocity round 

 its axis of symmetry, oz, which is also the axis of symmetry of the 

 material. The surface of the spheroid is supposed free of all forces. 



The terms depending on two only of the groups of associated con- 

 stants suffice, along with a particular solution on account of the 

 existence of what is equivalent to the occurrence of bodily forces, to 

 satisfy all the conditions of the problem, and the strains are deter- 

 mined explicitly. 



The limiting form of the solution when the polar axis of the 

 spheroid is supposed to diminish indefinitely, while the equatorial 

 remains unchanged, is applied to the case of a thin circular disk 

 rotating freely about a perpendicular to its plane through its centre. 

 The solution so obtained is shown to satisfy all the conditions required 

 for the circular disk, except that it brings in small tangential surface 

 stresses depending on terms of the order of the thickness of the disk. 

 According to this solution the disk increases in radius, and diminishes 

 everywhere in thickness, especially near the axis, so as to become 

 biconcave. All, originally plane, sections parallel to the faces become 

 very approximately paraboloids of revolution, the latus rectum of 

 each varying inversely as the square of the angular velocity into the 

 original distance of the section considered from the central section. 



Again, by supposing the ratio of the polar to the equatorial 

 diameter of the spheroid to become very great, a surface is obtained 

 which near the central plane, z = 0, of the spheroid differs very little 



