Sir George Greenhill and Dr Bennett, Non-Spinning Gyrostat 243 



The Rotation of the Non-Spinning Gyrostat. By Sir George 

 Greenhill and Dr G. T. Bennett. 



[Read 22 November 1920.] 



Dr G. T. Bennett has done me the honour of criticizing a state- 

 ment in § 14, p. 13, of my Report to the Aeronautical Committee on 

 Gyroscopic Theory, 1914 (cited as R. G. T.), and I am pleased to 

 have this opportunity of meeting his objections. 



The diagrams of Flatland on a sheet of paper are inadequate in 

 a discussion concerning Rotations in Space ; it is advisable then to 

 have a physical representation at hand, such as that described in 

 Fig. 3 of R. G. T., where the displacements can be visualised, with- 

 out the confusion arising from thinking of the different sides of 

 the sheet of paper of a diagram. 



In this model of Fig. 3, of an Altazimuth suspension of a stalk, 

 as an axle carrying a gyroscopic flywheel, the three angles 6,i/j,cf), 

 introduced into Dynamics by Euler in 1760, and standard to this 

 day in mathematical treatment, may receive the corresponding 

 astronomical names; 6, measured from the downward vertical Oz, 

 may be called the Nadir Distance; i/j will then be called the Azi- 

 muth ; and Euler's third angle </>, the angle the wheel has turned 

 over the axle OZ, may then be called the Hour Angle. 



The point in dispute is concerning this hour angle 0. 



On the model, of the stalk with altazimuth suspension, carrying 

 a flywheel moveable about the stalk as a smooth axle, the hour 

 angle (f> will represent the extent to which the wheel has rubbed 

 round relatively to the axle stalk. This displacement can be shown 

 unmistakably by chalk marks, originally in coincidence on wheel 

 and axle, and also on the frame, and in their subsequent divergence, 

 to show the increase in <f), and also in ijj. 



Dr G. T. Bennett will oblige greatly if he adheres to Euler's 

 standard notation; so I will change his equations (1), (2), (3) into 



deb ^dih ^ d(b dxjj ^difj da , , 



when the flywheel is not set spinning. Here da denotes the polar 

 element of area described on the unit sphere by Z the end of the 

 stalk; and, in an incomplete circuit, a must be taken to represent 

 the solid conical angle subtended by the spherical area bounded 

 by the arc described by Z, and the two great circles proceeding 

 from z to the ends of the arc. 



