254 H. A. Newton on the Motion of the Gyroscope, 



tliat of tlic lower side of tlie wheel, Tlie line E tlius describes 



a conical surface. 



The case in which this conical surface is that of a right cone, 

 and is described by a uniform velocity of O E, is the only one I 

 shall consider. It is proposed to show, that if such a motion is 

 produced by a force applied at E, that force must be constant, 

 and in the vertical plane P O E. 



For this purpose let the motion in question be considered as 

 generated in a different way. About E as a center, with some 

 radius P E, describe a circle in the plane A B C D. Through P 

 the highest point of this circle let a horizontal plane be drawn 

 cutting a vertical line through O in Z, join PZ. About Z as a 

 center describe in this plane a circle with P Z as radius. The 

 first of these two circles is in the wheel and moves with it. The 

 second is fixed in space. If the circumference of the first rolls 

 on that of the second, E will describe a right cone. 



Now by varying the ratio P E : P Z, we can vary at pleasure 

 the ratio of the velocity of rotation of the wheel about its axis 

 to that of the plane P E about Z. Any motion, therefore, 

 like that we have to consider, may be generated by the rolling of 

 one circle upon another. 



The point of contact P of the two circles is always in the same 

 vertical plane with OE. This point is for the instant at rest, and 

 the body may therefore be considered as revolving about OP.^ 



By this rotation the center of gravity E would describe a cir 



foreign 



centrifugal forces arising from rotation^^ nor 



The force at B must be such 



as Will chang 



This must be a downward force. For, the instant after P is 

 at rest P moves downward, and to produce this motion a force 

 in the plane POE is necessary. It must also be constant. For, 

 the circle which E tends to describe is always of the same radius^. 



Now besides foreign forces we must consider the a 

 forces arisinor from the rotation. The resultant of these 



The resultant of these must be 

 in the plane POE, since the body is symmetrical with respect 

 to that plane, and revolves about an axis in it. It is constant, 

 for the circumstances under which it is produced are ever the 

 same. It may be downward or upward. In either case, the 

 foreign force to be applied at E, being the sum or difference of 

 two con^stant forces, is itself constant. It must e^ddently be m 

 the plane of the other two, that is, in the vertical plane POE. 

 It requires then a constant downward force at E to produce a 

 uniform horizontal motion of the center of gravity. 



The meaning of this phrase may be explained by an example. When a wheel 

 revolves on an axis inclined to its plane, the rotation produces a tendency to in- 

 crease the angle of incIiDation. The causes of this tendencj are the so-called^^- 

 trifugal forces. They are treated as forces, becatige, in the motion of a free body 

 an oppomg force must be applied, if the axis of rotation is to remain unchanged. 



