90, 91] EQUATIONS OF TOP. 277 



The integral of energy 69) becomes 



F + f* -jT = -cos#, 



if we introduce the constants 



7 ox 2ft H* tMgl 



-A- AC> a = -r- 



The integral of vertical angular momentum 70) becomes 



2 

 ^4) sin # (p sin cp -f- q cos <p) = - ^ - = /5 6 cos #-, 



putting 



Inserting the values of p, q from 65), 



76) 



Eliminating - between the first two gives 



77) (ft -I cos #) 2 + sin 2 (^|) 2 = sin 2 (- cos ), 

 which if we put for cos# the single letter becomes simply 



T8 ) gf) = (i-^)(-^)-(^-^) 2 = /-(4 



From the second equation 76), 



and from the third, 



80) g-r-'-E#- 



The letter # represents the height above the origin of a point on 

 the axis of symmetry, at unit distance from the fixed point. This 

 point will be spoken of as the apex of the top. Equation 78) deter- 

 mines the rise and fall of the apex, equation 79) its horizontal motion. 



91. Top Equations deduced by Lagrauge's Method. 



Before proceeding with the discussion, let us find the equations by 

 means of Lagrange's method. We have the kinetic energy 



81) 



