93] REGULAR PRECESSION. SMALL OSCILLATIONS. 285 



Since throughout the motion 



we have the inequalities, 



110) C- dz > t > - - C- 



By means of a linear substitution we may simplify the integral. 

 Let us put 



/ 1 N Z 1~ Z Z 



111) 



when the integral becomes 



112) C dz - = C dx = p.na iff 4- 



J y^-ax*-*,) J yi-x*~ 



so that we have for t, 



113) cos"" 1 ^ > t + const. > cos" 1 ^. 



If now the difference 8 z% = x is sufficiently small in comparison 

 with 3 0J and # 3 ^ 2 , we may obtain an approximate result by 

 putting under the radical the mean of the quantities which are too 

 great and too small respectively, so that if ^ -f 2 2 = 2# we have 

 the approximate result 



114) 1 4- const. = cos- 1 x, 



from which we obtain 



x ^ c" " ~c = COS 



115) ^ = ^ + c cos 



The arbitrary constant has been taken so that i = when the top 

 is at its highest, and z = -f- c = r 



We thus see that when the roots z ly ^ are nearly enough equal, 

 the apex of the top rises and falls with a harmonic oscillation g of 



the small amplitude c = * ~ ' In order to determine when the 

 approximation is justified, we have to consider what will cause the 

 third root 3 to be large. Since # 2 and # 3 are the roots of the 

 quadratic function /i(#) 102), their sum is the negative of the coeffi- 

 cient of s divided by that of # 2 , that is 



a h 



