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



(6) ae'^-\-{d + b^We)4>"^-\- C {^\r'-{- <^' cos (9)2 = 



2h + (o^ [(<^ + c) cos^ 7 + 6 cos'7i + ^ ^05^72] 



the latter being the integral of kinetic energy. 



The relation between the angle /i and the latitude (X) of 

 the point ( 0) is expressed by the formula 



( 7 ) sin \ ■=■ cos a sin ^l + sin a cos /* cos yS . 



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

 through (0) and the center of the ring (C3), and yS the 

 angle of the planes XZ and Z 0^. 



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 /^ = -^ and, therefore, 



cos 7 = 0, cos 7j = sin dy cos 7^ = cos By 



(8) 2r = a6''2 + 6 (<^' + a))2sin2^ -^ d {<l>' + ayf 



+ C[^/r' + (<^' + a))cos(9]2. 



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



(9) ((^+6sin2 6')((/)'+a>) + C[A/r'+(<^'+a))cos6']cos6' = ?, 

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



(10) i/r' + ((/)' + ft)) cos ^ = ?i 



(11) a^'2+ {d-\-b sin^ 6) <^'2+ G (^/r' + </,' cos^)^ 



= 2/i + ft)2 (6 sin2(9+ Cqob^O) 



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

 may be presented thus : 



(9)' (cZ + 6 sin2 6') (<^' + a)) + CI, cos 6 = l^ 



