62 C. M. RROO^rALI. : 



than the above value the path described will be that of a pro- 

 late cycloid. At the moment of release the axis is moving 

 horizontally, but the gyroscopic force is not sufficient to retain 

 it in a horizontal path. It begins to fall, therefore, and it is 

 only after it has gained velocity by a considerable fall that the 

 gj^roscopic force is able to reverse the curvature and bring the 

 axis to horizontal motion. On the up half of the arc a similar 

 curve must be described in reverse order, as will be shown 

 below, and a uniform prolate cycloidal path therefore results, 

 ( Figure 3 ) . 



It is necessary at this point to digress a moment in order 

 to prove that, no matter what the circumstances of starting, 

 provided the apparatus is left thereafter to itself, the two 

 halves of each loop of the curve are of the same shape and 

 size. A process of reasoning similar to that used in the case 

 of starting from rest may be applied. Suppose the axis has 

 a certain velocity at starting. The deviating gyroscopic force 

 being always at right angles to the direction of motion, obvi- 

 ously cannot alter this initial velocity. The variations of 

 velocity due to the interchange of potential and kinetic energy 

 as the axis rises and falls under the dominance of the deviat- 

 ing force, are superposed upon this initial velocity. As before, 

 then, we have that at each level on either arc there is the same 

 velocity ( the initial velocity plus the velocity due to the ele- 

 vation ) and as a consequence the same deviating force. The 

 two halves of the curve, therefore, have equal radii at equal 

 elevations, a symmetrical condition of affairs that necessarily 

 makes both halves of the arc similar and equal. 



To resume : We have seen that if the apparatus is started 

 with a velocity less than a critical value, a prolate cycloid will 

 be described, the summits of which are at the original level. 

 If the axis is started with an initial velocity equal to the 

 critical value, a horizontal path results, as has also been 

 shown. If the apparatus is started with an initial velocity 

 greater than the critical value, the axis will at first rise, then 

 reverse curvature and finallv move horizontallv ; it will 



