J. G. Barnard on the ijryroscope. 69 



As the deflecting force of rotary motion is the sole agent in 

 diverting the vertical velocity produced by gravity from its 

 downward direction, and in producing these paradoxical effects; 

 and as the foregoing analysis while it has determined its value, 

 has thrown no light upon its origin, it may be well to inquire 



how this force is create 



Popular explanations have usually turned upon the deflexion 

 of the vertical components of rotary velocity by the vertical an- 

 gular motion of the axis produced by gravity. In point of fact, 

 however, hoth vertical and horizontal components are deflected, 

 one as much as the other ; and the simplest way of studying the 

 I effects produced, is to trace a vertical projection of the path of a 



point of the body under these combined motions. For this pur- 

 pose conceive the mass of the revolving disk concentrated in a 

 single ring of matter of a radius h due to its moment of inertia 

 G=Mk^^ (see Bartlett Mech. p. 178) andj for simplicity, suppose 

 the angular motion of the axis to take place around the centre 

 figure and of gravity (?. 



Let AB hid such a ring ^'^' ^• 



(supposed perpendicular to 

 the plane of projection) re- 

 volving about its axis of fig- 

 ure Cy while the axis turns 

 in the vertical plane about the 

 same point G. Let the rota- 

 tion be such that the visible 

 portion of the disk moves 

 upward through the semi-cir- 

 cumference, n-oni B to A^ 

 while the axis moves down- 

 ■ward through the angle d to 

 the position Q Q\ The point 

 B^ by its ojxial rotation alone, 



w^ould be carried to A ] but the plane of the disk, by simultane- 

 ous movement of the axis, is carried to the position A' B\ and 

 the point B arrives at B instead of A, through the curve pro- 

 jected in BGB\ The equation of the projection, in circular 

 functions, is easily made ; but its general character is readily 

 perceived, and it is sufficient to say, that it passes through the 

 point G^, — that its tangents at ^ and B' are perpendicular to AB 

 and A' B\ — and that its concavity, throughout its whole length, 

 turned to the right. The point A descends on the other, or re- 

 mote side of the disk, and makes an exactly similar curve AGA^ 

 with its concavity reversed. 



The centrifugal forces due to the deflections of the vertical 

 motions are normal to the concavities of these curves ; hence, on 

 the side of the axis towards the ej'e, they are to the hjl^ and on 



c 



I 



