eo 



/. G, Barnard on the Gyroscope^ 



point of it would describe an imdnlating curve (fig. 2) whose 

 superior culminations a, a\ ol\ &c., are cusps lying in tlie same 



Fig. 2. 



am 



tudes aa'j a' o!'^ &c.j as 



sin a 



n 



^ 



(? 



sin a : 71. If 



if evince of 



vaLiou « is uvj J iLiitj lULiu la Li^ uit aiamQLGT to me circui 



the circle : a property which, indicates the cycloid. 



Assuming a^OO"^ and sin «=!, equations (9) and (10) will give, 



by elimination of sm 



m^5J|^', 



~dt 



dt 



dip 



X 



2?Jr"' 



substitutin 



dw 



^ 



ndu 



V2 



45 



n 



differential equation of the 



2§ 



u — u 



diameter is 



1 



2p* 



In this position of the axis, both the angles u and ip are arcs 

 of great circles described by a point of the axis of figure at a 

 units distance from 0, and owing to their minuteness may be con- 

 sidered as rectilinear co-ordinates. 



If 



-^sin a; but then, while 



the angular motion ip is the same, the arc described by the same 

 point of the axis will beithat of a small circle^ whose actual 

 length will likewise be reduced in the ratio of l:sin«. The 

 curve is therefore a cycloid in all circumstances; and the axis of 

 figure moves as if it were attached to the circumference of a 



1 . 



-— sin a, which rolled along 



mi 



the horizontal circle, aa'a'\ about the vertical through the point 

 of support. 



The centre e of this little circle moves with uniform velocity- 

 The Jirst term of the value of V (equation 11) is due to this uni- 

 form motion : it may be called the mean precession. 



> 



