24 Trans. Acad. Sci. of St. Louis. 



(6) a^'2 + ((^ + &sm''^)</)'2+ (7(i^'+ <^'cos^)» = 



^ 2 /i + w''' [( a + c) cos^ 7 + 6 cos' 7^ + C cos^ 7,] 



the latter being the integral of kinetic energy. 



The relation between the angle yu, and the latitude (\) of 

 the point ( 0) is expressed by the formula 



(7) sin X = cos a sin /i* + sin « cos /x cos ^. 



a being the angle of the axis Z with the line ( 0^) passing 

 through (0) and the center of the ring (C3), and /8 the 

 angle of the planes XZ and ZO^. 



Suppose, now, that the axis ZZ' be made parallel to OH 

 and the mounting of the gyroscope then fixed in this position. 



In this case /* = ^ ^°^» therefore, 



cos 7 = 0, cos 7i = sin ^, cos 7^ = cos ^, 



(8) 2T = ae"^ + h {4>'-\-(oy&\n^d +d(</>' + «)a 



+ (7[^' + (f + a))cos6']2. 



We can, now, obtain a new integral, namely, 



(9) ((^+6sin2 6/)(<^'+ft))+C[^'+(0'+«)cos^]cos^ = Z, 

 while the integrals ( 5 ) and ( H ) take the form 



(10) i/r' + (</>' + 0)) cos <9 = Zi 



(11) a^"+ (cZ + 6 sin^ d) <^'>+ C7 (^/r' + <^' cos^)« 



= 2/^ + ft)' (6 sin2 d + C cos' ^) 



With the help of equation (10) the integrals (9) and (11) 

 may be presented thus : 



(9)' ((Z + 6sin2^) (^' + ^0) + CI, cos e = l^ 



