Chessin — On the Motion of Gyroscopes. 25 



2 



(7 _ (7/ COS 6') 

 (11)' a^'^ = ?3-- ^ + 6sm^r 



where we have put 



(12) ^3=2/i-6V + 2a)?,-tZco2 



From (11)' we find 



a I Vl,{. 



l/d+6sm2 6' dd 



(13) ^ = ±i/a I i/;^(rf4. 6sin2(9)— (?,— a;,cos^)2 



i 



Formula (13) shows that t is expressed in function of 6 by 

 a quadrature which involves, in general, hyperelliptic integ- 

 rals. This is the reason why in the treatment of the present 

 problem it has been customary to neglect the mass of the rings 

 ( (7i) and ( C^) of the gyroscope. In fact, if we put A^ = A^_ 

 = C7, = (7^ = 0, we shall have cZ = 0, 



■/ 



d 



sin Odd 



t = ztVab I ^ bi.sm^e — (I,— 01, cos ey 



i 



and the integration may be performed with the aid of circular 

 functions. But it is not at all necessary to perform the inte- 

 gration in order to get an idea of the character of the motion 

 in the general case, as we will now proceed to show. 



Let us put 



A = CH^ + bl,, a = 1//3 [(6 + d) A — 6/;^] , 

 (14)^ ^_ ^^A-^ n= ^lk±l. 



m 



