THE THEORY OF DIABOLO 569 



about its axis of symmetry, while a flat round thing has a 

 maximum moment of inertia about this axis. A bullet or an 

 ordinary diabolo spool is an instance of the former, while a 

 fly-wheel or a disc is an instance of the latter. A diabolo spool 

 need not have the usual angle of about 70 , it may be made 

 longer or wider. As it is made wider the moments of inertia in 

 the two directions become gradually more alike, and if it were 

 made wide enough it would correspond in this sense to a 

 fly-wheel, and the respective values would be inverted. There 

 must be, therefore, some intermediate width or angle of cone 

 which makes the moments of inertia in all directions the same, 

 so that dynamically it is the same as a sphere. 



It is easy to prove that in the case of a thin hollow double 

 cone, with the ends open and joined point to point, the proportion 

 which gives the spherical characteristic is reached when the angle 

 of the cone is nearly uo°, or when the tangent of the semi- 

 vertical angle is V2. All matter added to such an ideal skeleton, 

 without it equatorially makes the axis of rotational symmetry an 

 axis of maximum moment of inertia, all added beyond in the 

 direction of the axis makes this a minimum. A judicious 

 addition of matter without and beyond leaves the double cone 

 dynamically the same as a sphere. If, then, a spool is made 

 of this angle, and of such a thickness as to give a convenient 

 neck for the string by addition of matter without the ideal cone, 

 and by the addition of about the same amount beyond the ideal 

 cone, the resulting spool will be dynamically very nearly a 

 sphere. It is best to make it with an axial hole and with a 

 very slight preponderance of matter without the ideal cone. 

 The moment of inertia about the axis of spin is then just a 

 maximum, and the spool spins perfectly. A number of sticks 

 may then be made of different lengths and thrust centrally, 

 one at a time, into the hole. If any one of these is so exactly 

 chosen in length as to make the moments of inertia in all 

 directions the same, then spinning becomes impossible for the 

 reasons given. In order to ascertain if any stick is too long 

 or too short it is merely necessary to suspend the combination 

 from a torsion wire so as to oscillate about the axis of rotational 

 symmetry and about a transverse axis successively. Then if 

 the time of oscillation about the former is the greater, the stick 

 is too short, and vice versa. In order to make spinning really 

 hopeless the time of oscillation should not differ by anything 



