452 On the Residues of Powers of Numbers, fyc. [May 5, 



The linear momentum of the cyclic motion is also found to be 



A few cases of motion of the ring are then discussed. 



6. The annular form, of fluid rotating in relative equilibrium is 

 next considered, when the radius of the mean circle of the ring is at 

 least twice as great as the mean radius of the cross-section. 



The equation of the cross-section is assumed to be 



p = a (l + Pi cos % + cos2x + ), 



where p l9 /3 2 , &c, are small quantities. 



Taking the centre of gravity of the cross-section as origin, y3 x is 

 seen to be of the 5th order, and it is shown that, as far as the 4th 

 order, 



The shape of the cross-section is roughly elliptical, the major axis 

 of the ellipse being perpendicular to the axis of revolution. 



Y. " On the Residues of Powers of Numbers for any Com- 

 posite Modulus, Real or Complex." By Geoffrey T. 

 Bennett, B.A. Communicated by Professor Cayley, 

 F.R.S. Received April 8, 1892. 



(Abstract.) 



The present work consists of two parts, with an Appendix to the 

 second. Part I deals with real numbers, Part II with complex. 



In the simple cases, when the modulus is a real number which is 

 an odd prime, a power of an odd prime, or double the power of an 

 odd prime, we know that there exist primitive roots of the 



