Septejikee 25, 1896.] 



SCIENCE. 



445 



fast, and the tendency is to retain small 

 contracted figures. In both cases, however, 

 the occurrence of spirals is chiefly the re- 

 sult decreasing angular velocity due to 

 friction of the pencil on the tablet, as will 

 be presently shown. 



The most interesting results and by far 

 the most beautiful curves are obtained when 

 the tablet is not quite horizontal. In pro- 

 portion as larger angles of dip are chosen, 

 the spirals increase in lateralness until in an 

 extreme case they merge more and more 

 fully into contracting prolate cycloids. 

 The figures are in the main cornucopia-like 

 and the tracery at proper angles of dip is 



Fig. 2. 



exceedingly delicate. Figure 2 is a type 

 of these curves -'^ to which how:ever an in- 

 finite variety may be given, a dip of J to 1 

 inch in two feet being favorable. 



One would at first thought suppose that 

 a top on an inclined nearly smooth tablet 



* Unfortunately the photographic reproduction of 

 these curves was not satisfactory. The above figure 

 is made from a hand tracing of a coarsely drawn curve 

 and does not convey the finish of the originals. 



would tend to describe elongated figures by 

 sliding down the plane. Such, however, as 

 is otherwise known, is only in small part 

 the case. The top moves across the dip, 

 tending to remain, if not quite, at least very 

 nearly, on an average level. Moreover there 

 is a necessary relation between the dip, the 

 direction of rotation and the march of the 

 top across the dip. If the tablet slopes 

 downward from left to right parallel to the 

 observer, the top moves away from him if 

 spun counter-clockwise, and towards him if 

 spun clockwise. If the dip be downward 

 from right to left, the opposite relations of 

 rotations and progression will hold. In 

 other words, if the pivot or stylus of the top 

 were to point in the direction of the dip, 

 the rim or web would roll in the direction 

 in which the top actually moves across the 

 dip. 



The reason for this curious behavior 

 might perhaps most simply be looked for in 

 the fact that precession* is relatively less 

 accelerated when the end of the pencil 

 moves up hill and relatively more acceler- 

 ated when the end of the pencil rolls down 

 hill. Hence if the dip be from left to right 

 and the rotation counter-clockwise as seen 

 from above, the pencil sweeps further out 

 from the center, i. e., away from the ob- 

 server, because the obliquity of the top 

 axis is being relatively increased. In roll- 

 ing down hill the pencil sweeps nearer 

 towards the center of motion (^. e., also 

 away from the observer), because the ob- 

 liquity of the top axis is being relatively 

 decreased. An inspection of the experi- 

 ment and of the curves drawn by the top 

 does not bear this out. It appears rather 

 that the angular velocity of the top is con- 

 tinually decreased when the pencil rolls 

 up hill and is continually increased again 

 when the pencil rolls down hill. The tops 

 of the spires therefore correspond to a rela- 



* Following Lord Kelvin's well known explana- 

 tion. 



