378 ON THE MOTION OF 



If the body remains always nearly upright, these become 



U+My(^-gb»)+gy, = o, 



V-M 7 (^-ga")-gx, = o, 

 W = o. 



These equations furnish the same relations between F. G 

 and r as those obtained before. Either the rotation round the 

 natural vertical, or else those moments of inertia which would 

 (when made effective by a swift rotation) displace that ver- ' 

 tical, must be very small. If r is very small, the equations 

 of condition of perfect rolling are reduced to 



d£, = — yq&t, d>7, = ypdt . 



Substituting these values in equations (37), and recollecting 

 that £ -\-y = a", we shall find 



d£ = —a'qit, d>7, = a'pdt. 



But when r is small we have 



&% = d a £H-£dgrdf, d> a = d>,_ $dp&t. 



Therefore 



d a £ = — ydqdt, d> 3 = ydpdt; 



equations which are verified by the equation formerly obtain- 

 ed (37) when r is small, d% = d%, d\, = d\. Finally, 

 these last equations become, in consequence of the values 

 which p and q acquire when r is small, 



&% = ydV, d>. = yd'b"; 



