/. G. Bammed on the Gyroscope. 59 



extent of horizontal angular motion of the axis of figure after 

 any time tJ^ 



The first two will reach their respective maxima and minima 



when ^m^\^t=\ and —0; or when ^=^J— and <=?!-• 

 These values of t in equation (11) give 



n n 



Hence, counting from the commencement of motion when ?, u, 

 A -j-r and V' are all zero, we have the following series of corres- 



ponding values of these variables 



, ^ U 1 . dip \ \q 



• 4p^ 



2 



which correspond to the moment of greatest depression, when u 

 and j7 are maxima, and 



when, it appears (u being the zero), the axis of figure has re- 

 ained its original elevation and the horizontal velocity is 

 estroyed. 



All these values are (owing to the assumed large value of ^ 

 very minute. If we suppose the rotating velocity n = 100 ^-^ or 

 100 revolutions per second, the maximum of u (with an instru- 

 ment of ordinary proportions) would be a fraction of a minute 

 of arc, and the period of undulation but a fraction of a second. 

 Hence the horizontal motion about the point of support will 

 be exceedingly slow compared with the axial rotation of the disk 



1.^ expressed by n. 



If, in eq^uations (9) and (10), we increase t indefinitely, we will 

 find but a repetition of the series of Values already found, they 

 being recurring functions of the time. 



We see then the rcA'olving body noes not in fact maintain a 

 uniform unchanging elevation, and move about its point of sup- 

 port at a uniform rate, (as it appears to do). But the axis of fig- 

 * ure generates what may be called a corrugated cone^ and any 



w 



* The assumption that -d/r^O when t is zero supposes that the initial position of 

 the node coincides with the fixed axis of x. In my subsequent illustrations and 

 analysis T suppose the initial position to be at 90° therefroni, which would require 



to the above value of ^^, the constant - rt to be added. The horizontal angular 

 motion of the axis of fis-ure is the same as that of the node* 



